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

Newform orbit 58.6.a.a

Trace form

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

$2q + 8q^{2} - 14q^{3} + 32q^{4} - 82q^{5} - 56q^{6} - 160q^{7} + 128q^{8} - 344q^{9} - 328q^{10} - 502q^{11} - 224q^{12} - 186q^{13} - 640q^{14} + 134q^{15} + 512q^{16} - 1532q^{17} - 1376q^{18} - 2844q^{19} - 1312q^{20} + 2176q^{21} - 2008q^{22} + 2012q^{23} - 896q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{22})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 96x + 522$

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
$\operatorname{tr} a_p$	8	−14	−82	−160	−502	−186	−1532	−2844	2012	1682	6550	8008	−3680	17938	−2790	4218	12964	−79292	−45904	−53100	−140248	−105594	1140	7488	−124032

$\operatorname{tr} a_n$ 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)[0]   # 58.6.a.a

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

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