Atlas › Modular forms › Weight 4 › Level 58 › 58.4.a › 58.4.a.c

Newform orbit 58.4.a.c

Trace form

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

$2q - 4q^{2} - 2q^{3} + 8q^{4} - 10q^{5} + 4q^{6} - 16q^{7} - 16q^{8} - 40q^{9} + 20q^{10} - 90q^{11} - 8q^{12} - 50q^{13} + 32q^{14} - 62q^{15} + 32q^{16} - 44q^{17} + 80q^{18} + 108q^{19} - 40q^{20} + 112q^{21} + 180q^{22} - 28q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{6})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 12x - 114$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
29	$-1$

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	−2	−10	−16	−90	−50	−44	108	−28	58	66	40	304	−130	−514	−958	−180	1028	−912	796	−856	−318	−1828	−944	368

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 29, 58]: [1, 2, 9, 25] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(58, 4)[2]   # 58.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 58:4 1000