ex-delta

/*** EXAMPLE: Modular form Delta                                    ***/
/*** (illustration of unusual coefficient growth)                   ***/
/***                                                                ***/
/*** v1.2, July 2013, questions to [email protected]     ***/
/*** 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),")");