ex-eisen

/*** EXAMPLE: Eisenstein series G_k, k>2 even                       ***/
/*** v1.2, July 2013, questions to [email protected]     ***/
/***                                                                ***/
/*** type \rex-eisen or read("ex-eisen") at gp prompt to run this   ***/

read("computel");                 \\ read the ComputeL package
                                  \\ and set the default values
default(realprecision,40);        \\ set working precision; used throughout

K = 16;    \\ Our modular form is G_K with this K
           \\ may change this to any even K

conductor = 1;              \\ exponential factor
gammaV    = [0,1];          \\ list of gamma-factors
weight    = K;              \\ L(s)=sgn*L(weight-s)
sgn       = (-1)^(K/2);     \\ sign in the functional equation

\\ It has a simple pole in s=K
Lpoles    = [K];
Lresidues = [(-1)^(K/2)*sqrt(Pi)*bernfrac(K)/K];

initLdata("sigma(k,K-1)");  \\ initialize L-series 
                            \\ Coefficients given by the divisor function

print("EXAMPLE: L-function associated to the modular form G_",K," of weight ",K);
print("         coefficients = divisor function sigma(n,",K-1,")");
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),")");