Let $f_E$ be the newform attached to the Elliptic Curve $E$ with cremona label $\textbf{100a1}$ and let $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. Then $g = f[\alpha]_2 \in S_2(\Gamma_1(100))$. I recently computed the Fourier expansion of $g$ up to certain $q^N$, and it starts as: \begin{align} g(q) = \bigg(& -\frac{3}{5} \zeta_{10}^{2} + \frac{4}{5} \zeta_{10} - \frac{3}{5}\bigg)q + \bigg(\frac{8}{5} \zeta_{10}^{3} + \frac{6}{5} \zeta_{10} - \frac{6}{5}\bigg)q^{3} \\[0.1in] & + \bigg(-\frac{12}{5} \zeta_{10}^{3} + \frac{12}{5} \zeta_{10}^{2} + \frac{6}{5}\bigg)q^{5} + \bigg(\frac{6}{5} \zeta_{10}^{3} + \frac{2}{5} \zeta_{10}^{2} + \frac{6}{5} \zeta_{10}\bigg)q^{7} \\[0.1in] &+ \bigg(-\frac{1}{5} \zeta_{10}^{3} + \frac{4}{5} \zeta_{10}^{2} - \frac{4}{5} \zeta_{10} + \frac{1}{5}\bigg)q^{9}+ O(q^{11}), \end{align} where $\zeta_{10} = e^{\pi i/5}.$
I have some observations/questions regarding these $q$-expansions:
It turns out that the leading term $a_1(g)$ of $g$ is not an algebraic integer but $\frac{1}{a_1(g)}g$ seems to have coefficients in $\bar{\mathbb{Z}}$. Is this always the case for any newform at any cusp of width 1?
Also the computation suggests that $\mathbb{Q}(\{a_n(g)\}_n) = \mathbb{Q}(a_1(g)) \ (= \mathbb{Q}(\zeta_{10}))$ in this case. Does this generalize to newforms at width-1 cusps?
(minor) This computation is part of one research project of mine as a Ph.D. student. Just wondering if these computations have been done before and I'm reinventing the wheel, or it is of interest?
