Atlas › Modular forms › Weight 10 › Level 31 › 31.10.a › 31.10.a.b

Newform orbit 31.10.a.b

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 31q^{2} + 154q^{3} + 3925q^{4} + 1703q^{5} + 96q^{6} + 9109q^{7} + 54444q^{8} + 117909q^{9} - 143059q^{10} + 45994q^{11} - 75022q^{12} + 42242q^{13} + 289803q^{14} + 451126q^{15} + 2342433q^{16} + 832578q^{17} + 4018939q^{18} + 1992645q^{19} + 1817723q^{20} + 1754706q^{21} + 6215920q^{22} + 2677066q^{23} + 8489236q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 185x^{12} - 175168x^{11} + 24768405x^{10} + 11936200685x^{9} - 1067764522666x^{8} - 385899998175146x^{7} + 13058720908609236x^{6} + 5519273579377736152x^{5} + 94272585720179250160x^{4} - 20770430323470869998432x^{3} - 814793548973530859035968x^{2} - 2322731702795533516667520x + 142557653977616216030697216$

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 even 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
$\operatorname{tr} a_p$	31	154	1703	9109	45994	42242	832578	1992645	2677066	7613404	12005773	−9310718	36349541	−81554560	−10609144	122989656	44867703	297734646	541137644	1148060683	−26165066	796333276	570740118	549996964	−683459773

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 25] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 7 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, 10)[1]   # 31.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 31:10 1000