ECLIB with quadratic character
[latexpage]
ECLIB with quadratic character (with John Cremona; C++)
A modified version of John Cremona’s ECLIB libraries to compute weight 2 modular symbols for $\Gamma_0(N)$, now allowing a quadratic character $\chi$.
Status: in progress
Released: April 2011
Size: Modified ???? lines of code.
Availability: Google Code
License: GPL2
Other Information: Associated Documentation/Papers, Example Code
Extended Description:
This code modifies John Cremona’s well-established C++ code that computes modular symbols for congruence subgroups $\Gamma_0(N)$ to allow modular symbols with quadratic character $\chi$. This does not compute the much larger space of modular symbols for the subgroup $\Gamma_1(N)$ — instead we compute the twisted modular symbols by using Manin relations twisted by $\chi$ as described in Cremona’s paper [CITE THIS!], and we compute eigensymbols with a twisted Hecke action.
Modifications are done to store a quadratic Dirichlet character as a table of its values on $\mathbb{Z}/N\mathbb{Z}$, and to properly compute these twisted relations and actions.