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

Newform orbit 35.8.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 2q^{2} - 37q^{3} + 280q^{4} - 500q^{5} + 1144q^{6} - 1372q^{7} + 1512q^{8} + 2391q^{9} + 250q^{10} - 7225q^{11} + 28500q^{12} + 7957q^{13} + 686q^{14} + 4625q^{15} + 51288q^{16} + 64727q^{17} + 96526q^{18} - 37866q^{19} - 35000q^{20} + 12691q^{21} + 59056q^{22} + 111470q^{23} + 209264q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 39x^{3} - 6349x^{2} - 265597x - 2620702$

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
$\operatorname{tr} a_p$	−2	−37	−500	−1372	−7225	7957	64727	−37866	111470	218711	219348	−212844	−1215338	1074782	110699	3568154	261488	5355426	10352960	3607808	2787392	−735975	2001724	−13872714	−12763773

$\operatorname{tr} a_n$ 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)[2]   # 35.8.a.c

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

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