Atlas › Modular forms › Weight 10 › Level 7 › 7.10.a › 7.10.a.a

Newform orbit 7.10.a.a

Trace form

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

$2q - 6q^{2} - 86q^{3} - 620q^{4} - 2238q^{5} - 3988q^{6} - 4802q^{7} + 2616q^{8} + 11038q^{9} + 43384q^{10} + 35316q^{11} + 52136q^{12} - 26530q^{13} + 14406q^{14} - 307136q^{15} - 752q^{16} - 463920q^{17} + 332042q^{18} - 925426q^{19} + 473760q^{20} + 206486q^{21} + 1177888q^{22} + 778128q^{23} + 3301296q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{193})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 92x - 17184$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$

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
$\operatorname{tr} a_p$	−6	−86	−2238	−4802	35316	−26530	−463920	−925426	778128	−10003584	2467260	30735552	−19103448	4065100	−82195020	−55189812	−7069218	44316386	−241921336	206493816	−499153188	468535096	444023958	636267396	−1632716064

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.273 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, 7]: [1, 7] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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(7, 10)[0]   # 7.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 7:10 1000