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

Newform orbit 130.4.a.b

$q$ -expansion

$q + 2q^{2} - 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} - 8q^{7} + 8q^{8} - 11q^{9} - 10q^{10} - 32q^{11} - 16q^{12} - 13q^{13} - 16q^{14} + 20q^{15} + 16q^{16} - 86q^{17} - 22q^{18} - 56q^{19} - 20q^{20} + 32q^{21} - 64q^{22} + 68q^{23} - 32q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{7}$ is a root of

$x + 28$

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
$\operatorname{tr} a_p$	2	−4	−5	−8	−32	−13	−86	−56	68	−202	−56	66	490	460	−24	−294	−480	−338	676	120	−210	184	−660	−286	−1202

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 130, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[1]   # 130.4.a.b

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

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