Newform orbit 762.2.a.j

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$, whose minimal polynomial is

$x^{2} + 118x + 2501$

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 1.108 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(762, 2)[9]   # 762.2.a.j

cargo run --release -p sagebrush-web --example atlas -- 762:2 1000