## ECLIB with quadratic character

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** 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.