Atlas › Modular forms › Weight 4 › Level 62 › 62.4.a › 62.4.a.b

Newform orbit 62.4.a.b

$q$ -expansion

$q + 2q^{2} - 8q^{3} + 4q^{4} - 3q^{5} - 16q^{6} - 35q^{7} + 8q^{8} + 37q^{9} - 6q^{10} - 46q^{11} - 32q^{12} + 20q^{13} - 70q^{14} + 24q^{15} + 16q^{16} + 8q^{17} + 74q^{18} + 97q^{19} - 12q^{20} + 280q^{21} - 92q^{22} + 28q^{23} - 64q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 11$

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
$\operatorname{tr} a_p$	2	−8	−3	−35	−46	20	8	97	28	−206	−31	−282	367	−562	−148	−84	−301	−236	60	699	−814	670	−650	1566	−615

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 62, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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, 31, 62]: [1, 2, 9, 26] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(62, 4)[1]   # 62.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 62:4 1000