Function: mfmanin
Section: modular_forms
C-Name: mfmanin
Prototype: Gb
Help: mfmanin(FS): Given the modular symbol FS associated to an eigenform F
 by mfsymbol(mf,F), computes the even and odd special polynomials as well as
 the even and odd periods om+ and om- as a vector [[P+,P-],[om+,om-,r]],
 where r = imag(om+*conj(om-))/<F,F>.
 If F has several embeddings into C, give the vector of results corresponding
 to each embedding.
Doc: Given the modular symbol $FS$ associated to an eigenform $F$ by
 \kbd{mfsymbol(mf,F)}, computes the even and odd special polynomials as well
 as the even and odd periods $\omega^{+}$ and $\omega^{-}$ as a vector
 $[[P^{+},P^{-}],[\omega^{+},\omega^{-},r]]$, where
 $r=\Im(\omega^{+}\overline{\omega^{-}})/<F,F>$. If $F$ has several embeddings
 into $\C$, give the vector of results corresponding to each embedding.
 \bprog
 ? D=mfDelta(); mf=mfinit(D); DS=mfsymbol(mf,D);
 ? [pols,oms]=mfmanin(DS); pols
 %2 = [[4*x^9 - 25*x^7 + 42*x^5 - 25*x^3 + 4*x],\
       [-36*x^10 + 691*x^8 - 2073*x^6 + 2073*x^4 - 691*x^2 + 36]]
 ? oms
 %3 = [0.018538552324740326472516069364750571812,\
      -0.00033105361053212432521308691198949874026*I, 4096/691]
 ? mf=mfinit([11,2],0); F=mfeigenbasis(mf)[1]; FS=mfsymbol(mf,F);
 ? [pols,oms]=mfmanin(FS);pols
 %5 = [[0, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 0],\
       [2, 0, 10, 5, -5, -10, -10, -5, 5, 10, 0, -2]]
 ? oms[3]
 %6 = 24/5
 @eprog
