Atlas › Modular forms › Weight 4 › Level 35 › 35.4.a › 35.4.a.b

Newform orbit 35.4.a.b

Trace form

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

$2q + 8q^{2} + 2q^{3} + 20q^{4} - 10q^{5} - 8q^{6} - 14q^{7} + 48q^{8} + 12q^{9} - 40q^{10} - 14q^{11} - 108q^{12} + 50q^{13} - 56q^{14} - 10q^{15} + 168q^{16} - 50q^{17} + 16q^{18} + 36q^{19} - 100q^{20} - 14q^{21} - 184q^{22} + 244q^{23} - 496q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} - 10x + 7$

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
$\operatorname{tr} a_p$	8	2	−10	−14	−14	50	−50	36	244	−26	−120	564	−328	−260	−350	−56	−1232	336	−152	−1904	676	1014	−376	−216	2742

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 3, 3, 14] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 6 = 1 x 6 (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(35, 4)[1]   # 35.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 35:4 1000