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

Newform orbit 130.4.a.f

Trace form

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

$2q + 4q^{2} + 2q^{3} + 8q^{4} - 10q^{5} + 4q^{6} + 16q^{7} + 16q^{8} + 50q^{9} - 20q^{10} + 102q^{11} + 8q^{12} + 26q^{13} + 32q^{14} - 10q^{15} + 32q^{16} + 100q^{18} + 58q^{19} - 40q^{20} - 188q^{21} + 204q^{22} - 210q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{51})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 50x - 650$

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 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
$\operatorname{tr} a_p$	4	2	−10	16	102	26	0	58	−210	12	−242	−632	432	82	192	−240	942	460	−596	−990	832	388	−192	1380	−572

$\operatorname{tr} a_n$ 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)[5]   # 130.4.a.f

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

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