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

Newform orbit 35.8.a.a

Trace form

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

$2q + 16q^{2} - 30q^{3} - 40q^{4} + 250q^{5} - 768q^{6} - 686q^{7} - 960q^{8} - 756q^{9} + 2000q^{10} - 7906q^{11} - 7848q^{12} - 17818q^{13} - 5488q^{14} - 3750q^{15} - 4320q^{16} - 2398q^{17} + 9792q^{18} - 3612q^{19} - 5000q^{20} + 10290q^{21} - 96688q^{22} + 13844q^{23} + 24960q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{11})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 14x - 1051$

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 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
$\operatorname{tr} a_p$	16	−30	250	−686	−7906	−17818	−2398	−3612	13844	−126898	252768	−265860	−111920	947572	271274	−1267792	−1360120	−1813680	−2189312	−1494928	7169788	−7942974	−304712	−17943528	4258074

$\operatorname{tr} a_n$ 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)[0]   # 35.8.a.a

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

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