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

Newform orbit 62.4.a.a

$q$ -expansion

$q - 2q^{2} - 2q^{3} + 4q^{4} + q^{5} + 4q^{6} - 11q^{7} - 8q^{8} - 23q^{9} - 2q^{10} - 18q^{11} - 8q^{12} - 82q^{13} + 22q^{14} - 2q^{15} + 16q^{16} - 6q^{17} + 46q^{18} + 25q^{19} + 4q^{20} + 22q^{21} + 36q^{22} + 58q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

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

$x + 1$

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	−2	1	−11	−18	−82	−6	25	58	180	31	−146	47	−12	−136	−232	715	−518	−436	387	678	660	−382	−800	−1631

$\operatorname{tr} a_n$ 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)[0]   # 62.4.a.a

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

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