Some ambiguity in the output of mfslashexpansion

[Pisco Au <pisco125@hotmail.com>]

Hello,

I encountered some ambiguities when I'm computing $q$-expansion of modular forms using PARI's `mfslashexpansion`.

Let $\chi$ be a (complex-valued) Dirichlet character modulo 5 with order $4$. The space $M_1(\Gamma_0(5),\chi)$ has dimension $2$. The following computes its Fourier expansions:

    G = znstar(5,1); 

    L = [chi | chi<-chargalois(G), zncharisodd(G,chi)];

    mf = mfinit([5,1,[G,L[1]]]);

    B = mfbasis(mf);

    mfcoefs(B[1],10)

This says $M_1(\Gamma_0(5),\chi)$ is spanned by two forms, with $q$-expansions:

$$f_1 = \frac{3+i}{10} + q + (1+i)q^2 +(1-i)q^3 + O(q^4)$$

$$f_2 = \frac{3-i}{10} + q + (1-i)q^2 +(1+i)q^3 + O(q^4).$$

So far so good. Now I want to compute the $q$-expansion of $f_i$ at cusp $1/2$, so let $\gamma$ be the 2 by 2 matrix  $\left(\begin{smallmatrix}1&0 \\ 2& 1 \end{smallmatrix}\right)$, inputting 

    mfslashexpansion(mf,B[1],[1,0;2,1],3,0)

produces $$(0.137638 + 0.032492 i) + (-0.169572 - 0.413818 i)q^{1/5} + (0.328408 + 0.540507 i)q^{2/5} + (0.630313 + 0.0520147 i)q^{3/5} +O(q^{4/5})$$

This $q$-expansion, with explicit complex numbers, could be either $(f_1| \gamma)(\tau)$ or $(f_2| \gamma)(\tau)$. Because `B[1]` carries no preferences in embedding, which one of the above expansion equals to? $(f_1|_1\gamma)(\tau)$ or $(f_2|_1\gamma)(\tau)$?

Another confusion is when I type following, which computes the expansion using exact arithemetic:

    mfslashexpansion(mf,B[1],[1,0;2,1],3,1)

it returns `POLMOD` with polynomial $t^8 - t^6 + t^4 - t^2 + 1$. Since we have only two forms $f_1,f_2$, only two roots of this $8$-th degree equation actually occurs. How can I know which two? 

Finally, is it possible to directly compute $q$-expansion of $(f_1+f_2)|\gamma$? Note that $f_1+f_2$ has rational Q-coefficient. 

I have been looking at embedding of modular forms in PARI, but so far couldn't find appropriate way to do this. Thank you very much.

Best,

Kam Cheong Au