Atlas › Modular forms › Weight 8 › Level 62 › 62.8.a › 62.8.a.b

Newform orbit 62.8.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 32q^{2} - 68q^{3} + 256q^{4} - 334q^{5} - 544q^{6} - 948q^{7} + 2048q^{8} + 3016q^{9} - 2672q^{10} - 12618q^{11} - 4352q^{12} - 19760q^{13} - 7584q^{14} - 37338q^{15} + 16384q^{16} - 462q^{17} + 24128q^{18} - 42716q^{19} - 21376q^{20} - 106622q^{21} - 100944q^{22} - 132366q^{23} - 34816q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 402x^{3} - 35931x^{2} - 3522414x + 236938226$

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
$\operatorname{tr} a_p$	32	−68	−334	−948	−12618	−19760	−462	−42716	−132366	−198904	−119164	−198770	338250	307536	228464	−2260934	−1082416	−1764188	−729416	9797408	4740572	−2530394	15631746	−4628542	406622

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 3, 19, 58] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 6 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, 8)[1]   # 62.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 62:8 1000