Atlas › Modular forms › Weight 6 › Level 62 › 62.6.a › 62.6.a.a

Newform orbit 62.6.a.a

Trace form

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

$2q - 8q^{2} - 8q^{3} + 32q^{4} + 2q^{5} + 32q^{6} - 50q^{7} - 128q^{8} + 298q^{9} - 8q^{10} + 332q^{11} - 128q^{12} - 608q^{13} + 200q^{14} - 1512q^{15} + 512q^{16} - 3064q^{17} - 1192q^{18} - 1810q^{19} + 32q^{20} - 3560q^{21} - 1328q^{22} - 16q^{23} + 512q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{94})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 6x - 367$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
31	$+1$

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	−8	2	−50	332	−608	−3064	−1810	−16	−1492	−1922	−3908	16270	−23540	18712	−6168	−7174	31504	−79128	−13022	−65964	19868	169124	−4812	50450

$\operatorname{tr} a_n$ 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, 31, 62]: [1, 2, 15, 42] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; 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(62, 6)[0]   # 62.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 62:6 1000