Atlas › Modular forms › Weight 6 › Level 31 › 31.6.a › 31.6.a.a

Newform orbit 31.6.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 9q^{2} - 20q^{3} + 53q^{4} - 72q^{5} - 144q^{6} - 108q^{7} - 687q^{8} - 523q^{9} - 784q^{10} - 1238q^{11} - 712q^{12} - 166q^{13} - 486q^{14} + 602q^{15} + 2097q^{16} - 1284q^{17} + 4827q^{18} + 656q^{19} + 2442q^{20} - 550q^{21} + 7328q^{22} - 3994q^{23} + 8304q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 29x^{4} + 112x^{3} - 2974x^{2} - 26304x - 55456$

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
$\operatorname{tr} a_p$	−9	−20	−72	−108	−1238	−166	−1284	656	−3994	−7758	−4805	−17416	−27484	11600	−13976	−21292	38228	−49930	145356	22088	33590	−38702	−112666	−210820	−77120

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.217 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, 15] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 3 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, 6)[0]   # 31.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 31:6 1000