## Some computations of character sums

For a Dirichlet character $$\chi$$ modulo q, it is a theorem that the maximum of the character sum, $$M(\chi)$$, is always bigger than a constant times $$q^{1/2}$$ and it is also a theorem that the percentage of $$\chi$$ modulo q such that $$M(\chi) \lt \alpha q$$ goes to 0 as q goes to infinity. This is the motivation for the following data, which is related to a paper that Leo Goldmakher and I wrote.

First, here is a histogram of $$M(\chi)/q^{1/2}$$ for q = 1128931, with the data split into even and odd characters.

There is not much point to showing more histograms for q prime, really. They all look about the same. But you can see an animation of histograms for increasing primes.

Below are some tables of values of statistics for the distribution, split first into even and odd characters, and then with all characters. In the tables below, for each q, the minimum and maximum of $$M(\chi)$$ are listed along with some p-quantiles. The q's are a randomly chosen list of increasing primes.

Do these numbers approach a limit? (The maximum does not.)

If you like this, you might also enjoy:
• My paper with Leo Goldmakher.
• The LMFDB, which might contain this data some day.

### Even characters

 q min 0.2 0.5 0.9 0.99999 max 67 0.440488 0.478076 0.504441 0.587146 0.639042 0.639042 1069 0.440353 0.510355 0.542225 0.616498 0.791107 0.791107 2423 0.446221 0.509109 0.548597 0.625531 0.828182 0.828182 3023 0.442333 0.511584 0.547266 0.623894 0.831599 0.831599 4621 0.433619 0.514352 0.549275 0.625256 0.895405 0.895405 5417 0.441178 0.512637 0.549539 0.628284 0.856257 0.856257 6397 0.429469 0.513311 0.549741 0.627548 0.915757 0.915757 7321 0.441154 0.51286 0.55023 0.628165 0.900053 0.900053 8731 0.446683 0.514588 0.550127 0.628257 0.890133 0.890133 9059 0.450947 0.513656 0.551492 0.628371 0.90813 0.908130 10589 0.436819 0.514585 0.550362 0.62543 0.957579 0.957579 11909 0.437031 0.514304 0.551422 0.630495 0.962166 0.962166 12433 0.434252 0.515474 0.551269 0.630257 0.935498 0.935498 13513 0.434720 0.516462 0.552112 0.628969 0.966284 0.966284 14503 0.430981 0.515965 0.550936 0.629512 0.956104 0.956104 15803 0.430010 0.515152 0.551723 0.628904 0.895631 0.895631 16477 0.434144 0.514707 0.551265 0.628755 0.916274 0.916274 17207 0.428716 0.516044 0.551806 0.631011 0.944818 0.944818 18047 0.438850 0.515758 0.55202 0.632328 0.928463 0.928463 19267 0.423128 0.51559 0.551774 0.629886 0.93422 0.934220 20089 0.436064 0.51586 0.552649 0.630169 0.926881 0.926881 21799 0.426458 0.515549 0.552525 0.629787 0.929549 0.929549 22147 0.423835 0.516762 0.552088 0.628663 0.990308 0.990308 23099 0.430310 0.515786 0.552247 0.629348 0.935882 0.935882 24767 0.426694 0.516784 0.551686 0.631343 0.932642 0.932642 25589 0.436912 0.515068 0.552059 0.629792 0.938475 0.938475 26641 0.436726 0.516078 0.552321 0.629301 0.971034 0.971034 27259 0.439395 0.515915 0.552641 0.630178 0.988081 0.988081 28661 0.426519 0.5167 0.552913 0.629102 0.947911 0.947911 29021 0.422141 0.517555 0.552213 0.629967 0.918626 0.918626 30161 0.430778 0.516356 0.552612 0.630735 0.958806 0.958806 31333 0.431559 0.516207 0.552748 0.63001 0.99443 0.994430 32429 0.430291 0.516678 0.552435 0.630636 0.93416 0.934160 33469 0.426685 0.516473 0.552357 0.629477 0.959409 0.959409 34403 0.432499 0.517065 0.552733 0.631566 0.976206 0.976206 35491 0.427089 0.516317 0.553104 0.631458 0.957177 0.957177 36457 0.432674 0.515866 0.552433 0.62989 0.957056 0.957056 37397 0.436353 0.516445 0.55274 0.630233 0.968171 0.968171 38069 0.436671 0.516733 0.552095 0.630005 0.936402 0.936402 39383 0.429809 0.516603 0.552411 0.630074 0.966885 0.966885 40507 0.439864 0.516662 0.553037 0.630848 0.943715 0.943715 41233 0.438998 0.516576 0.553441 0.630018 0.985493 0.985493 42989 0.421131 0.516447 0.552842 0.631289 0.97994 0.979940 43051 0.437297 0.516001 0.552415 0.630734 0.978811 0.978811 44531 0.425204 0.516893 0.552957 0.631519 0.983143 0.983143 45139 0.432560 0.516344 0.552722 0.629071 0.98692 0.986920 46769 0.421788 0.516679 0.552917 0.630532 1.00588 1.005880 47591 0.424536 0.51653 0.552789 0.629713 1.00663 1.006627 48281 0.425722 0.516682 0.553016 0.630855 0.974865 0.974865 49123 0.421696 0.516839 0.552688 0.631441 0.968143 0.968143 50893 0.429161 0.517045 0.553081 0.630229 0.979828 0.979828 51307 0.431553 0.516749 0.552703 0.632117 0.951492 0.951492 52127 0.431166 0.516769 0.553003 0.631189 0.986425 0.986425 53267 0.438614 0.516874 0.553468 0.631184 0.992816 0.992816 54437 0.431416 0.516857 0.553395 0.629844 0.974256 0.974256 55609 0.414333 0.516663 0.553417 0.630906 0.990161 0.990161 56333 0.425837 0.517046 0.553122 0.631026 1.00394 1.003937 57751 0.432298 0.516534 0.553039 0.630512 1.00038 1.000383 58057 0.435011 0.516623 0.553023 0.630735 0.972773 0.972773 59141 0.426799 0.516887 0.552867 0.631565 0.977096 0.977096 60899 0.434179 0.516791 0.552877 0.630128 0.941461 0.941461 61357 0.423929 0.516728 0.553316 0.629921 0.951021 0.951021 62903 0.420265 0.516776 0.552902 0.631308 0.975027 0.975027 63901 0.430804 0.516759 0.552849 0.631167 1.02462 1.024621 64811 0.432101 0.516945 0.553099 0.631139 0.964088 0.964088 65407 0.425582 0.516718 0.552767 0.631443 0.960086 0.960086 66593 0.425393 0.516618 0.553155 0.630784 0.986872 0.986872 67751 0.423200 0.517234 0.553077 0.631017 0.98727 0.987270 68489 0.428231 0.517115 0.553199 0.632074 1.02261 1.022614 69473 0.430000 0.51698 0.553246 0.631232 1.00493 1.004928 70667 0.428418 0.51678 0.553341 0.630778 0.99487 0.994870 71789 0.421447 0.516997 0.553589 0.630677 1.04303 1.043033 72953 0.419186 0.517078 0.553229 0.631336 0.982263 0.982263 73847 0.425893 0.516956 0.553255 0.630977 1.00874 1.008743 74843 0.412579 0.517029 0.55298 0.631182 1.01513 1.015127 75709 0.431883 0.516726 0.553693 0.632192 1.03408 1.034083 76717 0.420029 0.516975 0.553145 0.631281 1.03333 1.033334 77317 0.426190 0.51717 0.553216 0.631289 0.998683 0.998683 78539 0.430674 0.517049 0.552855 0.630283 0.996383 0.996383 79939 0.427027 0.517297 0.55357 0.631138 0.992152 0.992152 80651 0.424633 0.517402 0.5532 0.631166 1.02518 1.025184 81611 0.433519 0.516974 0.553123 0.631032 0.988877 0.988877 82883 0.427615 0.517255 0.553295 0.631447 0.994875 0.994875 83591 0.415482 0.517373 0.553421 0.631689 1.01563 1.015629 84377 0.421204 0.517103 0.553262 0.630917 0.995922 0.995922 85643 0.432835 0.516981 0.553267 0.631737 0.969199 0.969199 86993 0.436081 0.517053 0.553064 0.631225 1.00327 1.003272 87181 0.431554 0.517222 0.55314 0.630902 0.981346 0.981346 88177 0.427035 0.517002 0.553386 0.630988 1.02864 1.028642 89783 0.417513 0.516898 0.553695 0.630587 1.01338 1.013379 90247 0.427056 0.517285 0.55324 0.631398 1.00004 1.000044 91757 0.421551 0.517001 0.553474 0.632061 1.03039 1.030392 92861 0.426740 0.517098 0.553397 0.631211 0.998514 0.998514 93683 0.424017 0.517029 0.553176 0.631211 0.993574 0.993574 94433 0.431929 0.517248 0.553494 0.63089 1.02517 1.025169 95857 0.425759 0.517168 0.553472 0.630996 1.001 1.001003 96601 0.431783 0.517218 0.55359 0.63106 1.01348 1.013484 97673 0.428917 0.51729 0.553197 0.631595 0.972075 0.972075 98597 0.423846 0.517214 0.553646 0.63166 0.970857 0.970857 99991 0.424659 0.517159 0.553482 0.631807 1.0111 1.011097 109841 0.423399 0.517346 0.553806 0.631336 0.994766 0.994766 119851 0.415567 0.517494 0.553614 0.630683 1.0574 1.057405 127931 0.419790 0.517421 0.553325 0.631493 1.03511 1.035114 133109 0.425388 0.517163 0.553699 0.631389 0.999952 0.999952 142403 0.417646 0.517569 0.553638 0.631511 1.00229 1.002288 152519 0.424360 0.517724 0.55352 0.632121 0.988631 0.988631 165887 0.420550 0.51765 0.553514 0.631585 0.995119 0.995119 177953 0.424707 0.517688 0.553886 0.631263 1.05724 1.057242 181213 0.426561 0.517482 0.55393 0.631376 1.01216 1.012161 197101 0.407015 0.517517 0.553768 0.631268 1.05951 1.059508 205937 0.429896 0.517343 0.553716 0.631497 1.01651 1.016510 210131 0.421151 0.51751 0.553965 0.631277 1.03823 1.038232 220009 0.417369 0.517735 0.553859 0.631582 1.04512 1.045122 237071 0.424736 0.517381 0.554124 0.631744 1.07061 1.070610 240733 0.426576 0.517556 0.553887 0.631462 1.04015 1.040155 259751 0.420162 0.517678 0.554125 0.632128 1.00343 1.003431 262331 0.421356 0.517529 0.554078 0.631721 1.00265 1.002652 279481 0.418743 0.517694 0.553921 0.631711 1.01721 1.017207 284311 0.414629 0.517749 0.554017 0.631699 1.03808 1.038083 295049 0.423653 0.51797 0.553859 0.631985 1.02507 1.025067 302297 0.414184 0.517723 0.554 0.63136 1.01762 1.017618 319577 0.417730 0.517576 0.554047 0.631654 1.03592 1.035919 326657 0.424214 0.517921 0.554062 0.632095 1.05582 1.055817 334753 0.421618 0.517743 0.554127 0.631309 1.0878 1.087800 343411 0.418871 0.517803 0.55403 0.632026 1.05649 1.056492 350617 0.422062 0.517842 0.554025 0.632232 1.06201 1.062013 362983 0.409468 0.517651 0.554081 0.631993 1.07959 1.079587 374083 0.423367 0.517729 0.554098 0.631762 1.04448 1.044481 381377 0.422006 0.517839 0.55387 0.631654 1.06064 1.060641 393989 0.420410 0.517818 0.554005 0.631457 1.03319 1.033187 404177 0.425492 0.517731 0.554075 0.631723 1.05093 1.080535 417883 0.417731 0.517856 0.553958 0.63205 1.03565 1.089701 425549 0.418261 0.517849 0.554162 0.631942 1.05081 1.070541 431803 0.423169 0.51776 0.554019 0.631873 1.05693 1.096593 445199 0.421707 0.517812 0.554126 0.632349 1.03927 1.044257 456499 0.419241 0.517909 0.554106 0.631695 1.02621 1.089081 467879 0.424806 0.51789 0.554094 0.632138 1.03975 1.046564 473833 0.409727 0.517847 0.553951 0.632041 1.05082 1.051418 481297 0.418793 0.517786 0.55422 0.631747 1.0341 1.051632 491501 0.416538 0.517811 0.554085 0.63195 1.05675 1.057152 506047 0.421180 0.51784 0.554034 0.632125 1.03052 1.050104 517873 0.423061 0.517975 0.554114 0.631977 1.06833 1.103451 527789 0.418630 0.517963 0.554158 0.632279 1.04409 1.078598 539783 0.421472 0.517756 0.554353 0.631838 1.04127 1.103634 540377 0.419038 0.517747 0.554185 0.632331 1.04428 1.050809 559907 0.423118 0.517929 0.554177 0.632095 1.0332 1.038355 562781 0.414868 0.517782 0.554098 0.631831 1.0221 1.096634 577939 0.404130 0.517828 0.554282 0.631824 1.04761 1.051409 585289 0.417024 0.517973 0.554158 0.631915 1.05131 1.076149 593783 0.419647 0.517991 0.554213 0.631609 1.03216 1.057647 607127 0.419903 0.517909 0.554304 0.631986 1.02696 1.029328 610523 0.420184 0.517893 0.554208 0.631962 1.06994 1.147833 620159 0.420258 0.51783 0.554456 0.631912 1.07271 1.086182 630871 0.416369 0.517868 0.554192 0.631922 1.06445 1.087675 641897 0.424171 0.517981 0.554184 0.631891 1.08182 1.111561 651103 0.421044 0.517939 0.554198 0.632125 1.04161 1.108804 667883 0.413223 0.517937 0.554199 0.632081 1.05855 1.071691 674299 0.420410 0.517793 0.554346 0.632214 1.06175 1.092752 686969 0.420646 0.518015 0.554304 0.631998 1.07658 1.076918 696853 0.415018 0.517965 0.554175 0.631697 1.06188 1.063630 700099 0.420635 0.517884 0.554242 0.63196 1.03843 1.073448 714503 0.403388 0.51804 0.554277 0.631943 1.07639 1.081860 721697 0.423175 0.517933 0.554109 0.632247 1.0555 1.066897 731249 0.416488 0.518047 0.554266 0.631653 1.07217 1.077550 746167 0.424757 0.517816 0.554381 0.632258 1.02195 1.063995 758893 0.420553 0.517915 0.554386 0.632178 1.05654 1.104019 769039 0.413051 0.517961 0.554436 0.631996 1.07106 1.080712 778313 0.419628 0.517915 0.554271 0.632051 1.03723 1.151143 788677 0.422306 0.517807 0.554316 0.632066 1.05127 1.059420 799171 0.422207 0.518108 0.554231 0.632167 1.06881 1.089774 802141 0.406761 0.517935 0.554241 0.632093 1.05991 1.104784 811523 0.413906 0.518055 0.554278 0.632166 1.05641 1.083153 820711 0.414216 0.517963 0.554252 0.63209 1.04225 1.044952 838589 0.422398 0.517979 0.554273 0.631942 1.05493 1.057199 844777 0.423870 0.517945 0.554309 0.63207 1.05196 1.096339 854129 0.421608 0.518023 0.554246 0.632188 1.05716 1.075656 865357 0.414149 0.517953 0.554266 0.631813 1.05512 1.092666 872129 0.401923 0.518047 0.554262 0.632109 1.11261 1.125619 885473 0.420575 0.518036 0.554347 0.631827 1.06167 1.093903 890887 0.421074 0.517877 0.554238 0.632387 1.05267 1.058165 908711 0.413265 0.517895 0.554265 0.631975 1.0542 1.100043 915053 0.419920 0.517919 0.554364 0.6319 1.06643 1.097321 922993 0.413538 0.517885 0.554359 0.632108 1.04357 1.058235 938057 0.419273 0.517961 0.554294 0.632011 1.05064 1.137302 943913 0.415958 0.517965 0.554368 0.632082 1.03429 1.076573 951341 0.412196 0.517924 0.554305 0.632125 1.0605 1.093042 964913 0.417622 0.518111 0.554274 0.632076 1.06835 1.086199 972991 0.421074 0.51793 0.554293 0.632351 1.0755 1.082571 984757 0.414311 0.517995 0.554438 0.632096 1.03152 1.129416 997307 0.423074 0.517902 0.554362 0.632215 1.06134 1.098660 1021403 0.420031 0.51799 0.554275 0.632119 1.05102 1.072589 1072129 0.418920 0.517885 0.554305 0.632403 1.06158 1.112955 1128931 0.419823 0.518014 0.554365 0.632226 1.0635 1.108900 1184587 0.412300 0.517991 0.554289 0.631997 1.05643 1.088526 1216951 0.419086 0.518064 0.554351 0.632164 1.03796 1.081835 1268173 0.415195 0.517988 0.55433 0.632133 1.04672 1.105893 1318063 0.422147 0.518038 0.554301 0.632128 1.04777 1.088708 1359997 0.414415 0.518041 0.554294 0.631993 1.04511 1.125557 1417057 0.424219 0.518172 0.554407 0.63225 1.06673 1.072026 1482937 0.417099 0.518012 0.554406 0.632287 1.06142 1.105539 1529149 0.417547 0.51809 0.55431 0.632168 1.07071 1.084897 1568971 0.416869 0.518111 0.554383 0.632248 1.08132 1.103758 1627481 0.413446 0.518081 0.554402 0.63211 1.0541 1.121146 1672063 0.409616 0.518084 0.554445 0.632199 1.05808 1.103044 1749073 0.421008 0.518198 0.554392 0.632151 1.07419 1.131459 q min 0.2 0.5 0.9 0.99999 max

### Odd characters

 q min 0.2 0.5 0.9 0.99999 max 67 0.551947 0.651235 0.733016 1.03923 1.09682 1.096815 1069 0.555453 0.679701 0.814012 1.16844 1.49621 1.496206 2423 0.489435 0.681224 0.81004 1.12313 1.57725 1.577248 3023 0.517474 0.691494 0.80742 1.11662 1.605 1.604996 4621 0.519704 0.692668 0.808513 1.12374 1.56429 1.564288 5417 0.503518 0.691093 0.808506 1.11067 1.56219 1.562188 6397 0.511638 0.693213 0.806981 1.11645 1.62483 1.624834 7321 0.491005 0.693526 0.811268 1.1187 1.68584 1.685842 8731 0.488295 0.689967 0.810002 1.12285 1.70295 1.702946 9059 0.499665 0.692506 0.808121 1.10446 1.71073 1.710727 10589 0.512070 0.694848 0.809528 1.10934 1.66953 1.669535 11909 0.463409 0.695686 0.808992 1.11388 1.61835 1.618351 12433 0.514707 0.693377 0.80765 1.12265 1.64383 1.643825 13513 0.494370 0.69431 0.813675 1.11639 1.71011 1.710111 14503 0.480125 0.6947 0.811204 1.11727 1.74797 1.747966 15803 0.493235 0.693918 0.811216 1.11227 1.66681 1.666806 16477 0.465137 0.692007 0.809298 1.11908 1.73743 1.737431 17207 0.483938 0.694739 0.809484 1.11779 1.74559 1.745589 18047 0.480087 0.694516 0.809266 1.11429 1.76183 1.761833 19267 0.490487 0.693582 0.810589 1.12131 1.69123 1.691234 20089 0.493049 0.693673 0.810905 1.12214 1.79316 1.793162 21799 0.471323 0.694914 0.811413 1.11964 1.75156 1.751556 22147 0.475735 0.694319 0.810712 1.12007 1.74552 1.745523 23099 0.489518 0.695183 0.81243 1.11957 1.77443 1.774429 24767 0.458282 0.694931 0.81058 1.11872 1.72683 1.726825 25589 0.472132 0.69545 0.812818 1.11861 1.75378 1.753785 26641 0.488856 0.696361 0.813227 1.1222 1.74598 1.745981 27259 0.474836 0.695212 0.811908 1.11936 1.77824 1.778236 28661 0.465423 0.695632 0.811922 1.12551 1.75164 1.751637 29021 0.499666 0.694094 0.811588 1.11855 1.76049 1.760486 30161 0.479579 0.694038 0.811294 1.12121 1.76773 1.767730 31333 0.490135 0.695232 0.812844 1.12327 1.85951 1.859513 32429 0.452990 0.696044 0.812311 1.12163 1.75678 1.756779 33469 0.491486 0.695916 0.811781 1.12015 1.72612 1.726121 34403 0.479100 0.695491 0.812753 1.12109 1.7918 1.791803 35491 0.478495 0.694933 0.811797 1.12046 1.72486 1.724861 36457 0.491001 0.693573 0.812128 1.12334 1.74521 1.745211 37397 0.476779 0.695015 0.811934 1.12013 1.8025 1.802503 38069 0.454236 0.694957 0.811768 1.12 1.75705 1.757046 39383 0.494332 0.694694 0.812466 1.11637 1.78074 1.780740 40507 0.494812 0.695811 0.811682 1.12039 1.75436 1.754365 41233 0.471819 0.695042 0.812088 1.12197 1.84008 1.840085 42989 0.481821 0.695555 0.81247 1.12147 1.79143 1.791433 43051 0.466717 0.694979 0.811976 1.12171 1.86392 1.863922 44531 0.476996 0.694246 0.812636 1.12207 1.79431 1.794305 45139 0.460749 0.696549 0.811777 1.12301 1.83347 1.833467 46769 0.484640 0.694599 0.811077 1.12129 1.84247 1.842472 47591 0.483616 0.695285 0.81168 1.12599 1.8183 1.818297 48281 0.463052 0.695049 0.812163 1.12311 1.84047 1.840466 49123 0.479781 0.695449 0.812028 1.12173 1.83958 1.839578 50893 0.447741 0.69542 0.8112 1.1246 1.7946 1.794604 51307 0.486939 0.696004 0.812183 1.12043 1.79265 1.792652 52127 0.474548 0.695466 0.811773 1.1192 1.82101 1.821011 53267 0.471767 0.694921 0.811771 1.12278 1.78028 1.780275 54437 0.466838 0.695405 0.809961 1.12048 1.86271 1.862706 55609 0.484018 0.694913 0.811948 1.1203 1.82336 1.823358 56333 0.460176 0.695714 0.812333 1.12378 1.87745 1.877450 57751 0.462555 0.694841 0.812741 1.11979 1.76809 1.768092 58057 0.484055 0.695111 0.813085 1.12297 1.86842 1.868422 59141 0.458880 0.696162 0.811734 1.12013 1.80541 1.805415 60899 0.477713 0.695119 0.811869 1.12546 1.92309 1.923094 61357 0.480208 0.695252 0.81342 1.12223 1.83564 1.835645 62903 0.469032 0.695211 0.81163 1.12236 1.83008 1.830079 63901 0.478489 0.695733 0.812966 1.12488 1.82256 1.822557 64811 0.486309 0.695002 0.811811 1.12327 1.80473 1.804730 65407 0.467137 0.694849 0.813139 1.12047 1.81764 1.817636 66593 0.476807 0.695895 0.811939 1.12621 1.78808 1.788081 67751 0.480013 0.69497 0.81249 1.12248 1.79614 1.796137 68489 0.472311 0.695829 0.812351 1.12351 1.82555 1.825554 69473 0.473451 0.695043 0.811749 1.12469 1.89264 1.892641 70667 0.481282 0.695216 0.812827 1.12286 1.85444 1.854436 71789 0.478961 0.695595 0.812483 1.12152 1.80165 1.801647 72953 0.478555 0.695983 0.812317 1.12377 1.82145 1.821451 73847 0.483041 0.696317 0.810885 1.12589 1.84904 1.849042 74843 0.485590 0.69442 0.813269 1.12747 1.88189 1.881887 75709 0.474644 0.6959 0.812073 1.12515 1.84522 1.845216 76717 0.478577 0.696414 0.811849 1.12223 1.89175 1.891753 77317 0.476432 0.69597 0.811967 1.12326 1.86145 1.861452 78539 0.494818 0.694743 0.813108 1.12556 1.87684 1.876835 79939 0.486693 0.695038 0.812772 1.12126 1.83591 1.835907 80651 0.467978 0.695347 0.812096 1.12125 1.84983 1.849835 81611 0.482579 0.695121 0.812646 1.12347 1.79276 1.792765 82883 0.475654 0.695735 0.81342 1.12286 1.82621 1.826211 83591 0.471938 0.695653 0.813589 1.12438 1.97841 1.978409 84377 0.480661 0.696574 0.812632 1.12494 1.80081 1.800812 85643 0.471025 0.695892 0.812396 1.12196 1.88632 1.886323 86993 0.481570 0.694566 0.812881 1.12388 1.80893 1.808926 87181 0.485704 0.695315 0.813028 1.12296 1.90978 1.909780 88177 0.473662 0.695961 0.813097 1.12453 1.85543 1.855432 89783 0.467464 0.696042 0.812559 1.12134 1.85729 1.857290 90247 0.466108 0.695773 0.813205 1.12373 1.86525 1.865254 91757 0.473394 0.696081 0.812103 1.12536 1.87354 1.873539 92861 0.464372 0.695709 0.812036 1.12389 1.84608 1.846083 93683 0.465518 0.695342 0.813301 1.1251 1.86591 1.865912 94433 0.466531 0.69506 0.812729 1.12495 1.87474 1.874737 95857 0.473569 0.695955 0.81286 1.12491 1.84302 1.843024 96601 0.473591 0.695644 0.812964 1.12338 1.86688 1.866882 97673 0.468513 0.696272 0.813304 1.12346 1.91103 1.911028 98597 0.485863 0.694977 0.812899 1.12485 1.89907 1.899068 99991 0.459954 0.69562 0.81263 1.12253 1.88727 1.887266 109841 0.478283 0.695933 0.812948 1.12535 1.82204 1.822041 119851 0.475207 0.695962 0.811975 1.12316 1.87028 1.870280 127931 0.479460 0.695675 0.813169 1.12474 1.85708 1.857079 133109 0.464575 0.695049 0.813008 1.1233 1.89217 1.892174 142403 0.468185 0.696356 0.813268 1.1249 1.90036 1.900359 152519 0.465777 0.69546 0.813444 1.1239 1.85972 1.859723 165887 0.470734 0.696509 0.812934 1.12614 1.92131 1.921314 177953 0.473030 0.696332 0.812989 1.12668 1.95138 1.951385 181213 0.467135 0.696149 0.812834 1.12456 1.91882 1.918819 197101 0.462055 0.696208 0.813095 1.1243 1.91461 1.914607 205937 0.462265 0.696174 0.813258 1.12451 1.89294 1.892937 210131 0.469373 0.695894 0.812837 1.12496 1.87536 1.875361 220009 0.448909 0.696041 0.813421 1.12331 1.87294 1.872941 237071 0.474138 0.696252 0.813681 1.125 1.92376 1.923763 240733 0.454017 0.696204 0.812932 1.12362 1.95896 1.958963 259751 0.459368 0.696327 0.813402 1.12354 1.935 1.935004 262331 0.475628 0.69703 0.813133 1.12401 1.92904 1.929039 279481 0.459708 0.69628 0.813139 1.12453 1.96764 1.967643 284311 0.464295 0.696074 0.813385 1.12438 1.92098 1.920983 295049 0.455635 0.696443 0.813129 1.12368 1.93957 1.939574 302297 0.462573 0.696397 0.812658 1.12461 1.8957 1.895704 319577 0.464159 0.696422 0.813241 1.12403 1.96537 1.965372 326657 0.444667 0.695995 0.813199 1.1255 1.86859 1.868589 334753 0.449366 0.696115 0.813211 1.12557 1.9539 1.953902 343411 0.462117 0.696281 0.813487 1.12454 1.93494 1.934937 350617 0.466506 0.696507 0.813174 1.12458 1.99342 1.993420 362983 0.449007 0.696534 0.813006 1.12433 1.93798 1.937978 374083 0.463335 0.696499 0.81344 1.12473 1.97431 1.974309 381377 0.452434 0.696263 0.813714 1.1246 1.9083 1.908296 393989 0.468544 0.696376 0.81359 1.12427 1.90988 1.909877 404177 0.444572 0.696277 0.81321 1.12488 1.95023 1.978918 417883 0.465483 0.69634 0.813 1.12474 1.95622 1.962479 425549 0.458719 0.696594 0.813396 1.12492 1.93453 1.942860 431803 0.467463 0.696327 0.813101 1.12459 1.91611 1.947089 445199 0.469737 0.696473 0.813619 1.1253 1.99872 2.114705 456499 0.469512 0.696855 0.813679 1.12551 1.93082 1.945206 467879 0.473381 0.696684 0.813252 1.12572 1.93319 1.951358 473833 0.470794 0.696408 0.813289 1.12488 1.8993 2.026617 481297 0.449198 0.69636 0.813334 1.12463 1.95464 1.987997 491501 0.469859 0.696246 0.813377 1.12425 1.9706 1.974505 506047 0.468243 0.697086 0.81322 1.12471 1.90809 1.980130 517873 0.456983 0.696395 0.813525 1.12454 1.94319 1.962408 527789 0.466454 0.696516 0.813363 1.12561 1.9129 1.966945 539783 0.464475 0.696192 0.813303 1.12437 1.93522 2.111067 540377 0.455210 0.696483 0.813222 1.12459 1.94555 1.992285 559907 0.445471 0.696163 0.813331 1.12618 1.92944 1.959241 562781 0.459730 0.696182 0.813522 1.12474 1.95764 2.014614 577939 0.452068 0.69668 0.813406 1.12532 1.94352 1.951873 585289 0.460026 0.696737 0.813369 1.12418 1.93946 1.944609 593783 0.443230 0.696212 0.813238 1.12465 1.94969 1.962575 607127 0.446276 0.696473 0.813428 1.12431 1.96585 1.979035 610523 0.471693 0.696573 0.813409 1.12539 1.90674 1.980025 620159 0.459177 0.696131 0.813361 1.12443 1.97651 2.024838 630871 0.461325 0.696596 0.813417 1.12444 1.97944 2.015696 641897 0.455330 0.696522 0.813499 1.12481 1.98878 1.993304 651103 0.468305 0.696221 0.813444 1.12515 1.95377 1.960255 667883 0.463317 0.696597 0.813429 1.12551 1.95122 1.979122 674299 0.460298 0.696331 0.813324 1.12499 1.97517 2.016545 686969 0.463408 0.69645 0.813386 1.12448 1.93961 1.940106 696853 0.450553 0.696452 0.813425 1.12514 1.99842 2.023095 700099 0.462714 0.696847 0.813327 1.12559 1.93904 1.949704 714503 0.462758 0.696557 0.813581 1.12488 1.95281 1.996721 721697 0.467575 0.696695 0.813345 1.12481 1.9936 2.003108 731249 0.459495 0.696335 0.813784 1.12522 1.96851 2.060999 746167 0.460657 0.696669 0.813568 1.12486 2.02752 2.031288 758893 0.450727 0.69641 0.813329 1.12491 1.99056 1.996820 769039 0.463119 0.696382 0.813475 1.12438 1.99967 2.009363 778313 0.455580 0.696582 0.813314 1.12473 2.00479 2.070913 788677 0.454112 0.696439 0.813688 1.12577 1.96859 1.969801 799171 0.467846 0.696576 0.813623 1.12468 2.01231 2.013125 802141 0.468286 0.696561 0.813661 1.1245 1.97638 2.003317 811523 0.456622 0.696611 0.813238 1.12417 1.95445 2.016184 820711 0.467086 0.696497 0.813398 1.12446 1.9987 2.053454 838589 0.456510 0.696701 0.81363 1.12498 1.92959 1.988603 844777 0.460028 0.696656 0.813516 1.12445 1.95373 2.007182 854129 0.460667 0.696473 0.813515 1.12432 1.97546 2.034326 865357 0.438522 0.696604 0.813339 1.12502 1.97589 2.024348 872129 0.458349 0.696615 0.813595 1.12454 1.94936 2.014314 885473 0.463396 0.696595 0.813496 1.12504 1.95039 2.041939 890887 0.461849 0.696459 0.813751 1.12524 1.97868 2.038182 908711 0.457774 0.696447 0.813649 1.12568 1.96282 2.002164 915053 0.465819 0.696595 0.813578 1.12468 1.98264 2.013886 922993 0.460311 0.696769 0.813617 1.12492 1.97311 2.034047 938057 0.456058 0.696749 0.813783 1.12474 1.97128 2.028083 943913 0.456500 0.696642 0.813414 1.12471 1.98292 2.145594 951341 0.465605 0.696478 0.813544 1.12491 1.98299 2.017978 964913 0.450342 0.696682 0.813457 1.12443 1.98477 2.020526 972991 0.463888 0.696724 0.813141 1.12491 1.95626 1.995796 984757 0.454095 0.696513 0.813599 1.12443 1.94835 1.959237 997307 0.448603 0.69655 0.813526 1.12497 1.95548 1.980562 1021403 0.462080 0.696565 0.813759 1.12479 1.96026 2.009609 1072129 0.452125 0.696517 0.813486 1.12504 1.9764 2.077659 1128931 0.459772 0.696481 0.813782 1.12523 1.98461 2.007457 1184587 0.451827 0.696627 0.813542 1.12461 1.97377 2.011848 1216951 0.454405 0.696576 0.813614 1.12475 1.98786 2.030872 1268173 0.464917 0.696727 0.813496 1.12499 1.93541 1.957919 1318063 0.462975 0.696716 0.813691 1.12469 1.99561 2.033223 1359997 0.458846 0.696631 0.81364 1.12444 1.98739 2.003792 1417057 0.453987 0.696826 0.813494 1.12459 1.96651 2.068574 1482937 0.450833 0.696775 0.813375 1.12493 1.97834 2.123144 1529149 0.462625 0.696846 0.813502 1.12429 2.03858 2.103055 1568971 0.456185 0.696592 0.813472 1.12479 1.98065 2.060433 1627481 0.458968 0.696665 0.813666 1.12493 1.94745 2.030444 1672063 0.459313 0.696643 0.813773 1.12525 1.95669 2.125299 1749073 0.446327 0.696919 0.813689 1.12478 1.99372 2.041590 q min 0.2 0.5 0.9 0.99999 max

### All characters

 q min 0.2 0.5 0.9 0.99999 max 67 0.440488 0.50273 0.598735 0.953201 1.09682 1.096815 1069 0.440353 0.530447 0.629641 0.980472 1.49621 1.496206 2423 0.446221 0.534163 0.63508 0.983659 1.57725 1.577248 3023 0.442333 0.533976 0.634047 0.986087 1.605 1.604996 4621 0.433619 0.536173 0.636086 0.984139 1.56429 1.564288 5417 0.441178 0.536082 0.638557 0.995652 1.56219 1.562188 6397 0.429469 0.538061 0.639842 0.995146 1.62483 1.624834 7321 0.441154 0.538731 0.638644 0.99133 1.68584 1.685842 8731 0.446683 0.537691 0.638705 0.989301 1.70295 1.702946 9059 0.450947 0.53831 0.641255 0.997223 1.71073 1.710727 10589 0.436819 0.539483 0.639084 0.997551 1.66953 1.669535 11909 0.437031 0.539191 0.639195 0.993361 1.61835 1.618351 12433 0.434252 0.538188 0.641349 0.99095 1.64383 1.643825 13513 0.434720 0.539496 0.639877 0.99764 1.71011 1.710111 14503 0.430981 0.539307 0.641258 0.991664 1.74797 1.747966 15803 0.430010 0.539633 0.639815 0.994441 1.66681 1.666806 16477 0.434144 0.539361 0.640111 0.99464 1.73743 1.737431 17207 0.428716 0.539194 0.642534 0.990364 1.74559 1.745589 18047 0.438850 0.539603 0.641988 0.996004 1.76183 1.761833 19267 0.423128 0.539618 0.641126 0.997279 1.69123 1.691234 20089 0.436064 0.540152 0.641172 0.99496 1.79316 1.793162 21799 0.426458 0.539792 0.641756 0.998106 1.75156 1.751556 22147 0.423835 0.540102 0.641365 0.998416 1.74552 1.745523 23099 0.430310 0.539593 0.640742 0.998885 1.77443 1.774429 24767 0.426694 0.538938 0.640804 0.995574 1.72683 1.726825 25589 0.436912 0.539905 0.640823 0.995861 1.75378 1.753785 26641 0.436726 0.539841 0.640442 0.995348 1.74598 1.745981 27259 0.439395 0.539708 0.640468 0.995615 1.77824 1.778236 28661 0.426519 0.54037 0.641429 0.994378 1.75164 1.751637 29021 0.422141 0.539942 0.64067 0.996237 1.76049 1.760486 30161 0.430778 0.539824 0.640691 0.995342 1.76773 1.767730 31333 0.431559 0.540171 0.641155 0.99549 1.85951 1.859513 32429 0.430291 0.540198 0.64113 1.00227 1.75678 1.756779 33469 0.426685 0.539923 0.639986 0.997278 1.72612 1.726121 34403 0.432499 0.540353 0.642121 0.998133 1.7918 1.791803 35491 0.427089 0.540363 0.641489 0.999154 1.72486 1.724861 36457 0.432674 0.540475 0.641834 0.999906 1.74521 1.745211 37397 0.436353 0.540411 0.641317 0.996219 1.8025 1.802503 38069 0.436671 0.540431 0.640779 0.996514 1.75705 1.757046 39383 0.429809 0.540418 0.641509 0.999269 1.78074 1.780740 40507 0.439864 0.540601 0.642435 0.998346 1.75436 1.754365 41233 0.438998 0.540234 0.641098 0.997864 1.84008 1.840085 42989 0.421131 0.540348 0.64152 0.996427 1.79143 1.791433 43051 0.437297 0.539747 0.641647 0.998249 1.86392 1.863922 44531 0.425204 0.540176 0.641704 0.997382 1.79431 1.794305 45139 0.432560 0.540742 0.642377 0.995606 1.83347 1.833467 46769 0.421788 0.540541 0.641542 0.996857 1.84247 1.842472 47591 0.424536 0.540617 0.641309 0.994942 1.8183 1.818297 48281 0.425722 0.540391 0.641022 0.997982 1.84047 1.840466 49123 0.421696 0.540791 0.641788 0.997304 1.83958 1.839578 50893 0.429161 0.541237 0.641683 0.996389 1.7946 1.794604 51307 0.431553 0.540294 0.641394 0.997021 1.79265 1.792652 52127 0.431166 0.540778 0.641761 0.996999 1.82101 1.821011 53267 0.438614 0.540919 0.643198 0.997107 1.78028 1.780275 54437 0.431416 0.541017 0.641956 0.996708 1.86271 1.862706 55609 0.414333 0.54123 0.642622 0.999104 1.82336 1.823358 56333 0.425837 0.54082 0.641503 0.996656 1.87745 1.877450 57751 0.432298 0.540974 0.641844 0.995575 1.76809 1.768092 58057 0.435011 0.540868 0.642628 0.99809 1.86842 1.868422 59141 0.426799 0.540807 0.641526 0.998071 1.80541 1.805415 60899 0.434179 0.541148 0.642193 0.9967 1.92309 1.923094 61357 0.423929 0.540337 0.642094 0.998889 1.83564 1.835645 62903 0.420265 0.540641 0.641722 0.997484 1.83008 1.830079 63901 0.430804 0.540808 0.641875 0.996455 1.82256 1.822557 64811 0.432101 0.540543 0.642124 0.996299 1.80473 1.804730 65407 0.425582 0.54045 0.641683 0.998235 1.81764 1.817636 66593 0.425393 0.540784 0.64146 0.995932 1.78808 1.788081 67751 0.423200 0.540984 0.642463 0.997079 1.79614 1.796137 68489 0.428231 0.540789 0.642832 0.997395 1.82555 1.825554 69473 0.430000 0.540797 0.642418 0.998585 1.89264 1.892641 70667 0.428418 0.54081 0.642271 0.995794 1.85444 1.854436 71789 0.421447 0.541387 0.642198 0.99828 1.80165 1.801647 72953 0.419186 0.540704 0.642382 0.99695 1.82145 1.821451 73847 0.425893 0.540901 0.642216 0.997759 1.84904 1.849042 74843 0.412579 0.540945 0.642419 0.99738 1.88189 1.881887 75709 0.431883 0.541336 0.642344 0.996503 1.84522 1.845216 76717 0.420029 0.541035 0.641837 0.99811 1.89175 1.891753 77317 0.426190 0.54123 0.642374 0.996191 1.86145 1.861452 78539 0.430674 0.541011 0.641522 0.996723 1.87684 1.876835 79939 0.427027 0.541139 0.641697 0.996776 1.83591 1.835907 80651 0.424633 0.541074 0.64223 0.998084 1.84983 1.849835 81611 0.433519 0.540914 0.642323 0.996338 1.79276 1.792765 82883 0.427615 0.540796 0.641953 0.997557 1.82621 1.826211 83591 0.415482 0.540591 0.642452 0.997567 1.97841 1.978409 84377 0.421204 0.541003 0.641723 0.998054 1.80081 1.800812 85643 0.432835 0.54082 0.641992 0.997571 1.88632 1.886323 86993 0.436081 0.540804 0.642578 0.998625 1.80893 1.808926 87181 0.431554 0.541295 0.642781 0.999025 1.90978 1.909780 88177 0.427035 0.540818 0.64204 0.998328 1.85543 1.855432 89783 0.417513 0.541124 0.641069 0.998226 1.85729 1.857290 90247 0.427056 0.54066 0.642134 0.997551 1.86525 1.865254 91757 0.421551 0.540984 0.642732 0.999263 1.87354 1.873539 92861 0.426740 0.540943 0.642103 0.998273 1.84608 1.846083 93683 0.424017 0.540864 0.642295 0.997627 1.86591 1.865912 94433 0.431929 0.540968 0.642256 0.997612 1.87474 1.874737 95857 0.425759 0.541313 0.642085 0.998151 1.84302 1.843024 96601 0.431783 0.541051 0.642343 0.99886 1.86688 1.866882 97673 0.428917 0.54106 0.642204 0.997434 1.91103 1.911028 98597 0.423846 0.541178 0.64256 0.996558 1.89907 1.899068 99991 0.424659 0.541215 0.64212 0.997863 1.88727 1.887266 109841 0.423399 0.541165 0.642075 0.997131 1.82204 1.822041 119851 0.415567 0.541331 0.642273 0.997751 1.87028 1.870280 127931 0.419790 0.541315 0.642295 0.998702 1.85708 1.857079 133109 0.425388 0.541057 0.64297 0.998594 1.89217 1.892174 142403 0.417646 0.541406 0.642381 0.998427 1.90036 1.900359 152519 0.424360 0.541185 0.64317 0.998058 1.85972 1.859723 165887 0.420550 0.541629 0.642385 0.998409 1.92131 1.921314 177953 0.424707 0.54147 0.642513 0.997341 1.95138 1.951385 181213 0.426561 0.541605 0.642582 0.997598 1.91882 1.918819 197101 0.407015 0.54144 0.642755 0.999377 1.91461 1.914607 205937 0.429896 0.541283 0.642515 0.998103 1.87427 1.892937 210131 0.421151 0.541818 0.642503 0.998101 1.85945 1.875361 220009 0.417369 0.54159 0.642212 0.997896 1.87245 1.872941 237071 0.424736 0.541576 0.642882 0.99815 1.89577 1.923763 240733 0.426576 0.541508 0.642513 0.997847 1.88857 1.958963 259751 0.420162 0.541717 0.642868 0.997521 1.89004 1.935004 262331 0.421356 0.54167 0.64271 0.998419 1.90868 1.929039 279481 0.418743 0.541704 0.642634 0.99804 1.92271 1.967643 284311 0.414629 0.541729 0.642698 0.99848 1.86811 1.920983 295049 0.423653 0.541575 0.642652 0.998257 1.93006 1.939574 302297 0.414184 0.541711 0.642607 0.998634 1.86727 1.895704 319577 0.417730 0.541823 0.64278 0.998799 1.91248 1.965372 326657 0.424214 0.541709 0.642963 0.998051 1.86514 1.868589 334753 0.421618 0.541892 0.642595 0.997484 1.91018 1.953902 343411 0.418871 0.541642 0.642819 0.997823 1.89919 1.934937 350617 0.422062 0.541635 0.643093 0.998594 1.91721 1.993420 362983 0.409468 0.541742 0.643129 0.998512 1.91352 1.937978 374083 0.423367 0.54179 0.642935 0.9988 1.93925 1.974309 381377 0.422006 0.541602 0.643022 0.998266 1.8913 1.908296 393989 0.420410 0.541813 0.642719 0.998793 1.88567 1.909877 404177 0.425492 0.541741 0.642722 0.998421 1.89328 1.978918 417883 0.417731 0.54172 0.643071 0.998457 1.90893 1.962479 425549 0.418261 0.541753 0.642869 0.998183 1.934 1.942860 431803 0.423169 0.541755 0.642833 0.998047 1.87108 1.947089 445199 0.421707 0.54178 0.643132 0.999255 1.98817 2.114705 456499 0.419241 0.541746 0.642807 0.997639 1.9256 1.945206 467879 0.424806 0.541842 0.642834 0.99804 1.92884 1.951358 473833 0.409727 0.541692 0.64297 0.998612 1.8924 2.026617 481297 0.418793 0.541801 0.642811 0.998121 1.94895 1.987997 491501 0.416538 0.541732 0.643125 0.99855 1.92777 1.974505 506047 0.421180 0.541707 0.643332 0.998547 1.90684 1.980130 517873 0.423061 0.541863 0.643216 0.998354 1.9077 1.962408 527789 0.418630 0.541835 0.642804 0.998266 1.90339 1.966945 539783 0.421472 0.542024 0.642901 0.998239 1.92719 2.111067 540377 0.419038 0.54182 0.643318 0.998198 1.93555 1.992285 559907 0.423118 0.541791 0.643123 0.997938 1.9294 1.959241 562781 0.414868 0.541838 0.642954 0.998245 1.94549 2.014614 577939 0.404130 0.541866 0.642964 0.998755 1.93526 1.951873 585289 0.417024 0.542023 0.643106 0.998645 1.93421 1.944609 593783 0.419647 0.541944 0.642809 0.998699 1.92971 1.962575 607127 0.419903 0.541807 0.642964 0.998621 1.9505 1.979035 610523 0.420184 0.541804 0.643002 0.998246 1.89111 1.980025 620159 0.420258 0.542038 0.64324 0.999541 1.93039 2.024838 630871 0.416369 0.541831 0.642731 0.998606 1.96878 2.015696 641897 0.424171 0.541826 0.643018 0.998465 1.91452 1.993304 651103 0.421044 0.541915 0.642856 0.998187 1.93327 1.960255 667883 0.413223 0.541895 0.642782 0.998118 1.93227 1.979122 674299 0.420410 0.541922 0.64308 0.998088 1.94571 2.016545 686969 0.420646 0.54182 0.643014 0.998895 1.91778 1.940106 696853 0.415018 0.541859 0.642999 0.998398 1.95242 2.023095 700099 0.420635 0.541868 0.642923 0.998335 1.91754 1.949704 714503 0.403388 0.541816 0.643104 0.998383 1.9472 1.996721 721697 0.423175 0.541903 0.643184 0.998474 1.96252 2.003108 731249 0.416488 0.541921 0.642965 0.997959 1.95068 2.060999 746167 0.424757 0.541961 0.643131 0.998386 1.97013 2.031288 758893 0.420553 0.541908 0.643123 0.998557 1.93806 1.996820 769039 0.413051 0.541888 0.643191 0.998483 1.97237 2.009363 778313 0.419628 0.541969 0.643197 0.99846 1.9209 2.070913 788677 0.422306 0.541953 0.643173 0.99824 1.92354 1.969801 799171 0.422207 0.541892 0.642922 0.998661 1.98126 2.013125 802141 0.406761 0.541896 0.643029 0.998378 1.95379 2.003317 811523 0.413906 0.542004 0.643084 0.997899 1.9357 2.016184 820711 0.414216 0.541837 0.643107 0.998776 1.91722 2.053454 838589 0.422398 0.541827 0.643093 0.998717 1.92253 1.988603 844777 0.423870 0.541906 0.643078 0.998981 1.9487 2.007182 854129 0.421608 0.542002 0.643217 0.998712 1.95028 2.034326 865357 0.414149 0.541904 0.643117 0.998651 1.95686 2.024348 872129 0.401923 0.541951 0.643101 0.998724 1.94042 2.014314 885473 0.420575 0.541969 0.642949 0.998449 1.91499 2.041939 890887 0.421074 0.541927 0.643304 0.998411 1.93288 2.038182 908711 0.413265 0.54186 0.643078 0.998135 1.93826 2.002164 915053 0.419920 0.541954 0.643119 0.998605 1.96467 2.013886 922993 0.413538 0.542011 0.643017 0.998985 1.96415 2.034047 938057 0.419273 0.541895 0.643004 0.998984 1.94439 2.028083 943913 0.415958 0.542026 0.643106 0.998223 1.9618 2.145594 951341 0.412196 0.542004 0.643129 0.998626 1.95906 2.017978 964913 0.417622 0.541933 0.643066 0.998588 1.97971 2.020526 972991 0.421074 0.542002 0.64337 0.998752 1.95506 1.995796 984757 0.414311 0.541947 0.643227 0.998478 1.93484 1.959237 997307 0.423074 0.54196 0.643254 0.998518 1.92335 1.980562 1021403 0.420031 0.541872 0.643143 0.998683 1.95367 2.009609 1072129 0.418920 0.541973 0.643179 0.998387 1.94659 2.077659 1128931 0.419823 0.541933 0.643303 0.998586 1.97573 2.007457 1184587 0.412300 0.542038 0.643058 0.998248 1.92327 2.011848 1216951 0.419086 0.541954 0.643378 0.99857 1.95418 2.030872 1268173 0.415195 0.542042 0.643182 0.998419 1.92766 1.957919 1318063 0.422147 0.541896 0.643031 0.998489 1.9742 2.033223 1359997 0.414415 0.541969 0.643169 0.998695 1.97241 2.003792 1417057 0.424219 0.542112 0.643092 0.998961 1.9524 2.068574 1482937 0.417099 0.542047 0.643063 0.998559 1.94502 2.123144 1529149 0.417547 0.542089 0.643263 0.998612 1.94918 2.103055 1568971 0.416869 0.541988 0.643065 0.998494 1.95952 2.060433 1627481 0.413446 0.54203 0.643199 0.998637 1.94178 2.030444 1672063 0.409616 0.542073 0.643253 0.998525 1.94117 2.125299 1749073 0.421008 0.542036 0.643301 0.998662 1.98845 2.041590 q min 0.2 0.5 0.9 0.99999 max