/*** EXAMPLE: Modular form Delta ***/ /*** (illustration of unusual coefficient growth) ***/ /*** ***/ /*** v1.2, July 2013, questions to tim.dokchitser@bristol.ac.uk ***/ /*** type \rex-delta or read("ex-delta") at pari prompt to run this ***/ read("computel"); \\ read the ComputeL package \\ and set the default values default(realprecision,40); \\ set working precision; used throughout \\ * Coefficients are given by Ramanujan's tau function \\ * Re-define the default bound on the coefficients: \\ Deligne's estimate on tau(n) is better than the default \\ coefgrow(n)=(4n)^(11/2) (by a factor 1024), \\ so re-defining coefgrow() improves efficiency (slightly faster) coefgrow(n) = 2*n^(11/2); tau(n) = (5*sigma(n,3)+7*sigma(n,5))*n/12\ -35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5)); \\ initialize L-function parameters conductor = 1; \\ exponential factor gammaV = [0,1]; \\ list of gamma-factors weight = 12; \\ L(s)=sgn*L(weight-s) sgn = 1; \\ sign in the functional equation initLdata("tau(k)"); \\ L-series coefficients a(k) print("EXAMPLE: L-function associated to the modular form Delta of weight 12"); print(" coefficients = Ramanujan's tau function"); print(" with ",default(realprecision)," digits precision"); print("Verifying functional equation. Error: ",errprint(checkfeq())); print("L(1) = ",lval = L(1)); print(" (check) = ",lval2 = L(1,1.1)," (err=",errprint(lval-lval2),")");