Atlas › Modular forms › Weight 8 › Level 35 › 35.8.a › 35.8.a.d

Newform orbit 35.8.a.d

Trace form

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

$5q + 11q^{2} + 65q^{3} + 553q^{4} + 625q^{5} + 96q^{6} + 1715q^{7} - 1647q^{8} - 12q^{9} + 1375q^{10} + 10027q^{11} + 10024q^{12} + 16141q^{13} + 3773q^{14} + 8125q^{15} + 94169q^{16} + 17427q^{17} + 47547q^{18} + 44394q^{19} + 69125q^{20} + 22295q^{21} - 89168q^{22} + 55698q^{23} + 18324q^{24} + O(q^{25})$

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

$x^{5} - 76x^{4} - 3266x^{3} + 227230x^{2} - 625391x - 43857426$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
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 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
$\operatorname{tr} a_p$	11	65	625	1715	10027	16141	17427	44394	55698	−164219	138440	181518	−369676	386	962985	396244	6284948	1975828	−554376	6526040	7479766	4851305	−11813196	12879896	11704767

$\operatorname{tr} a_n$ 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, 5, 7, 35]: [1, 5, 5, 30] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(35, 8)[3]   # 35.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 35:8 1000