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

Newform orbit 62.6.a.b

Trace form

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

$2q + 8q^{2} - 8q^{3} + 32q^{4} - 110q^{5} - 32q^{6} - 126q^{7} + 128q^{8} - 398q^{9} - 440q^{10} - 396q^{11} - 128q^{12} - 632q^{13} - 504q^{14} + 328q^{15} + 512q^{16} - 720q^{17} - 1592q^{18} - 438q^{19} - 1760q^{20} - 896q^{21} - 1584q^{22} - 560q^{23} - 512q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{7})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 118x + 3453$

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	−110	−126	−396	−632	−720	−438	−560	4404	1922	−8100	1742	−8572	26792	33656	9478	26328	−60224	−152490	−32860	−163276	−130324	−39908	11034

$\operatorname{tr} a_n$ 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)[1]   # 62.6.a.b

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

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