Atlas › Modular forms › Weight 4 › Level 31 › 31.4.a › 31.4.a.a

Newform orbit 31.4.a.a

Trace form

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

$2q - 5q^{2} - 2q^{3} + 5q^{4} - 25q^{5} - 12q^{6} - 19q^{7} - 15q^{8} - 18q^{9} + 88q^{10} - 26q^{11} + 80q^{12} + 64q^{13} + 90q^{14} - 26q^{15} + 57q^{16} - 84q^{17} + 79q^{18} - 51q^{19} - 190q^{20} - 66q^{21} + 116q^{22} + 10q^{23} - 104q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 7x + 8$

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
$\operatorname{tr} a_p$	−5	−2	−25	−19	−26	64	−84	−51	10	−314	62	32	−583	220	512	−54	−609	372	424	−317	−1840	−1132	1458	418	1363

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.091 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, 9] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 1 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, 4)[0]   # 31.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 31:4 1000