Atlas › Modular forms › Weight 4 › Level 130 › 130.4.a › 130.4.a.c

Newform orbit 130.4.a.c

Trace form

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

$2q - 4q^{2} - 8q^{3} + 8q^{4} + 10q^{5} + 16q^{6} - 20q^{7} - 16q^{8} + 38q^{9} - 20q^{10} + 44q^{11} - 32q^{12} - 26q^{13} + 40q^{14} - 40q^{15} + 32q^{16} - 92q^{17} - 76q^{18} - 132q^{19} + 40q^{20} - 160q^{21} - 88q^{22} - 336q^{23} + 64q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{30})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{2} + 68x - 2474$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$-1$
13	$+1$

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
$\operatorname{tr} a_p$	−4	−8	10	−20	44	−26	−92	−132	−336	−8	−180	272	140	208	−1164	−356	−412	88	−60	260	1136	−368	540	−148	−452

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.134 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, 2, 5, 10, 13, 26, 65, 130]: [1, 2, 3, 7, 5, 13, 24, 68] (certified at ell = 2147483629)
T = [(3, 1), (7, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(130, 4)[2]   # 130.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 130:4 1000