Function: _header_modular_symbols
Class: header
Section: modular_symbols
Doc:
 \section{Modular symbols}

 Let $\Delta_{0} := \text{Div}^{0}(\P^{1}(\Q))$ be the abelian group of
 divisors of degree $0$ on the rational projective line. The standard
 $\text{GL}(2,\Q)$ action on $\P^{1}(\Q)$ via homographies naturally extends to
 $\Delta_{0}$. Given

 \item $G$ a finite index subgroup of $\text{SL}(2,\Z)$,

 \item a field $F$ and a finite dimensional representation $V/F$ of
   $\text{GL}(2,\Q)$,

 \noindent we consider the space of \emph{modular symbols} $M :=
 \Hom_{G}(\Delta_{0}, V)$. This finite dimensional $F$-vector
 space is a $G$-module, canonically isomorphic to $H^{1}_{c}(X(G), V)$,
 and allows to compute modular forms for $G$.

 Currently, we only support the groups $\Gamma_{0}(N)$ ($N > 0$ an integer)
 and the representations $V_{k} = \Q[X,Y]_{k-2}$ ($k \geq 2$ an integer) over
 $\Q$. We represent a space of modular symbols by an \var{ms} structure,
 created by the function \tet{msinit}. It encodes basic data attached to the
 space: chosen $\Z[G]$-generators $(g_{i})$ for $\Delta_{0}$
 (and relations among
 those) and an $F$-basis of $M$. A modular symbol $s$ is thus given either in
 terms of this fixed basis, or as a collection of values $s(g_{i})$
 satisfying certain relations.

 A subspace of $M$ (e.g. the cuspidal or Eisenstein subspaces, the new or
 old modular symbols, etc.) is given by a structure allowing quick projection
 and restriction of linear operators; its first component is a matrix whose
 columns  form  an $F$-basis  of the subspace.
