Function: mstooms
Section: modular_symbols
C-Name: mstooms
Prototype: GG
Help: mstooms(Mp, phi): given Mp from mspadicinit, lift the
 (classical) eigen symbol phi to a distribution-valued overconvergent symbol
 in the sense of Pollack and Stevens.
 The resulting overconvergent eigensymbol can then be used in
 mspadicmoments, then mspadicL or mspadicseries.
Doc: given \kbd{Mp} from \kbd{mspadicinit}, lift the (classical) eigen symbol
 \kbd{phi} to a $p$-adic distribution-valued overconvergent symbol in the
 sense of Pollack and Stevens. More precisely, let $\phi$ belong to the space
 $W$ of modular symbols of level $N$, $v_p(N) \leq 1$, and weight $k$ which is
 an eigenvector for the Hecke operator $T_N(p)$ for a non-zero eigenvalue
 $a_p$ and let $N_0 = \text{lcm}(N,p)$.

 Under the action of $T_{N_0}(p)$, $\phi$ generates a subspace $W_\phi$ of
 dimension $1$ (if $p\mid N$) or $2$ (if $p$ does not divide $N$) in the
 space of modular symbols of level $N_0$.

 Let $V_p=[p,0;0,1]$ and $C_p=[a_p,p^{k-1};-1,0]$.
 When $p \not\mid N$ and $a_p$ is divisible by $p$, \kbd{mstooms}
 returns the lift $\Phi$ of $(\phi,\phi|_k V_p)$ such that
  $$T_{N_0}(p) \Phi = C_p \Phi$$

 When $p \not\mid N$ and $a_p$ is not divisible by $p$, \kbd{mstooms}
 returns the lift $\Phi$ of $\phi - \alpha^{-1} \phi|_k V_p$
 which is an eigenvector of $T_{N_0}(p)$ for the unit eigenvalue
 where $\alpha^2 - a_p \alpha + p^{k-1}=0$.

 The resulting overconvergent eigensymbol can then be used in
 \tet{mspadicmoments}, then \tet{mspadicL} or \tet{mspadicseries}.
 \bprog
 ? M = msinit(3,6, 1);
 ? xpm = [5,-3,-1]~;
 ? msissymbol(M, xpm)
 %3 = 1
 ? p = 5; mshecke(M,p) * xpm  \\ eigenvector of T_5, a_5 = 6
 %4 = [30, -18, -6]~
 ? Mp = mspadicinit(M, p, 10, 0); \\ restrict to ordinary symbols, mod p^10
 ? PHI = mstooms(Mp, xpm);
 ? PHId = mspadicmoments(Mp, PHI);
 ? mspadicL(PHId)
 %8 = 5 + 2*5^2 + 2*5^3 + ...
 @eprog
