Newform orbit 910.2.a.n

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$; the sum of their $q$-expansions is

$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 6q^{9} + 2q^{10} - 2q^{11} + 2q^{12} + 2q^{13} + 2q^{14} - 2q^{15} + 2q^{16} + 8q^{17} - 6q^{18} - 6q^{19} - 2q^{20} - 2q^{21} + 2q^{22} + 6q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{11} + 5T_{17} + 7T_{19} + 9T_{23} + 11T_{29}$, whose minimal polynomial is

$x^{2} - 92x + 2096$

Atkin–Lehner signs

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$. The Fricke sign is $w_N = -1$, so the sign of the functional equation is $(-1)^{k/2} w_N = +1$: each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.251 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

newform_orbits(910, 2)[13]   # 910.2.a.n

cargo run --release -p sagebrush-web --example atlas -- 910:2 1000