Atlas › Modular forms › Weight 6 › Level 58 › 58.6.a › 58.6.a.c

Newform orbit 58.6.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 16q^{2} - 13q^{3} + 64q^{4} - 7q^{5} + 52q^{6} - 74q^{7} - 256q^{8} + 779q^{9} + 28q^{10} - 483q^{11} - 208q^{12} - 433q^{13} + 296q^{14} - 4579q^{15} + 1024q^{16} - 4414q^{17} - 3116q^{18} - 4630q^{19} - 112q^{20} - 6370q^{21} + 1932q^{22} - 4416q^{23} + 832q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 20x^{3} - 3096x^{2} - 120192x - 1082880$

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
$\operatorname{tr} a_p$	−16	−13	−7	−74	−483	−433	−4414	−4630	−4416	−3364	5053	18778	−23680	−22547	−38767	353	−16920	−50398	−55984	−36474	184390	169699	−131964	860	−25588

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 13, 39] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 4 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, 6)[2]   # 58.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 58:6 1000