Atlas › Modular forms › Weight 8 › Level 31 › 31.8.a › 31.8.a.a

Newform orbit 31.8.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 17q^{2} - 14q^{3} + 229q^{4} - 430q^{5} - 528q^{6} - 832q^{7} - 135q^{8} + 2115q^{9} - 5232q^{10} - 7886q^{11} - 24952q^{12} - 21844q^{13} - 37902q^{14} - 37796q^{15} - 49583q^{16} - 54822q^{17} - 57869q^{18} - 45352q^{19} - 50110q^{20} - 90756q^{21} + 30696q^{22} - 18464q^{23} + 84472q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 31x^{6} - 8266x^{5} - 132930x^{4} + 15588628x^{3} + 111486640x^{2} - 8449179712x - 4467019008$

Atkin–Lehner signs

$p$	$w_p$
31	$-1$

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 odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−17	−14	−430	−832	−7886	−21844	−54822	−45352	−18464	−81488	208537	431648	−1465990	−598714	−2003572	−1496844	−2853828	−1486900	−5647492	−5168828	4710926	4582796	−626514	−13906634	2962898

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.266 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 engine's checks

dim M^+ at levels [1, 31]: [1, 19] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(31, 8)[0]   # 31.8.a.a

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 31:8 1000