Atlas › Modular forms › Weight 6 › Level 82 › 82.6.a › 82.6.a.d

Newform orbit 82.6.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 24q^{2} + 8q^{3} + 96q^{4} + 124q^{5} + 32q^{6} + 38q^{7} + 384q^{8} + 830q^{9} + 496q^{10} + 1262q^{11} + 128q^{12} + 1600q^{13} + 152q^{14} + 1278q^{15} + 1536q^{16} + 912q^{17} + 3320q^{18} + 776q^{19} + 1984q^{20} + 4980q^{21} + 5048q^{22} + 5456q^{23} + 512q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 132x^{5} - 9022x^{4} + 1740714x^{3} - 54907538x^{2} - 110043672x + 611151084$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
41	$+1$

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 even 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
$\operatorname{tr} a_p$	24	8	124	38	1262	1600	912	776	5456	620	−52	11008	−10086	−11228	−11896	−35600	−17248	−79612	−26758	−28112	60964	−101856	13204	106272	300052

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.256 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, 41, 82]: [1, 2, 18, 54] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(82, 6)[3]   # 82.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 82:6 1000