msfromcusp

Function: msfromcusp
Section: modular_symbols
C-Name: msfromcusp
Prototype: GG
Help: msfromcusp(M, c): returns the modular symbol attached to the cusp
 c, where M is a modular symbol space of level N.
Doc: returns the modular symbol attached to the cusp
 $c$, where $M$ is a modular symbol space of level $N$, attached to
 $G = \Gamma_0(N)$. The cusp $c$ in $\P^1(\Q)/G$ is given either as \kbd{oo}
 ($=(1:0)$) or as a rational number $a/b$ ($=(a:b)$). The attached symbol maps
 the path $[b] - [a] \in \text{Div}^0 (\P^1(\Q))$ to $E_c(b) - E_c(a)$, where
 $E_c(r)$ is $0$ when $r \neq c$ and $X^{k-2} \mid \gamma_r$ otherwise, where
 $\gamma_r \cdot r = (1:0)$. These symbols span the Eisenstein subspace
 of $M$.
 \bprog
 ? M = msinit(2,8);  \\  M_8(Gamma_0(2))
 ? E =  mseisenstein(M);
 ? E[1] \\ two-dimensional
 %3 =
 [0 -10]

 [0 -15]

 [0  -3]

 [1   0]

 ? s = msfromcusp(M,oo)
 %4 = [0, 0, 0, 1]~
 ? mseval(M, s)
 %5 = [1, 0]
 ? s = msfromcusp(M,1)
 %6 = [-5/16, -15/32, -3/32, 0]~
 ? mseval(M,s)
 %7 = [-x^6, -6*x^5 - 15*x^4 - 20*x^3 - 15*x^2 - 6*x - 1]
 @eprog
 In case $M$ was initialized with a nonzero \emph{sign}, the symbol is given
 in terms of the fixed basis of the whole symbol space, not the $+$ or $-$
 part (to which it need not belong).
 \bprog
 ? M = msinit(2,8, 1);  \\  M_8(Gamma_0(2))^+
 ? E =  mseisenstein(M);
 ? E[1] \\ still two-dimensional, in a smaller space
 %3 =
 [ 0 -10]

 [ 0   3]

 [-1   0]

 ? s = msfromcusp(M,oo) \\ in terms of the basis for M_8(Gamma_0(2)) !
 %4 = [0, 0, 0, 1]~
 ? mseval(M, s) \\ same symbol as before
 %5 = [1, 0]
 @eprog