Atlas › Modular forms › Weight 2 › Level 631 › 631.2.a › 631.2.a.b

Newform orbit 631.2.a.b

Trace form

The 32 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$32q + 7q^{2} + 2q^{3} + 33q^{4} + 17q^{5} + 11q^{6} - 2q^{7} + 21q^{8} + 52q^{9} + 5q^{10} + 11q^{11} + 2q^{12} + 9q^{13} + 12q^{14} - 12q^{15} + 27q^{16} + 39q^{17} + 12q^{18} + 9q^{19} + 31q^{20} + 25q^{21} - 11q^{22} + 4q^{23} + 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 32. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{32} - 9x^{31} - 93x^{30} + 1022x^{29} + 3122x^{28} - 48925x^{27} - 36651x^{26} + 1313568x^{25} - 400703x^{24} - 22224534x^{23} + 18740853x^{22} + 252376990x^{21} - 277215099x^{20} - 1998633274x^{19} + 2318645073x^{18} + 11289254743x^{17} - 12083838607x^{16} - 45830558540x^{15} + 39550647138x^{14} + 132248465893x^{13} - 76830700853x^{12} - 261633414220x^{11} + 72171389579x^{10} + 330817104732x^{9} + 2522785569x^{8} - 235526517316x^{7} - 59194270830x^{6} + 69303667601x^{5} + 33287114129x^{4} + 1747936018x^{3} - 939383427x^{2} - 128572656x - 3567193$

Atkin–Lehner signs

$p$	$w_p$
631	$-1$

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
$\operatorname{tr} a_p$	7	2	17	−2	11	9	39	9	4	39	2	0	121	−6	28	−2	59	7	−14	23	6	−29	41	131	2

$\operatorname{tr} a_n$ 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 631]: [0, 53] (certified at ell = 2147478481)
T = [(2, 1), (3, 1)]; h of degree 52 = 1 x 52 (phi(ord eps) x dim S_k^new); 6 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(631, 2)[1]   # 631.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 631:2 1000