Atlas › Modular forms › Weight 4 › Level 82 › 82.4.a › 82.4.a.c

Newform orbit 82.4.a.c

Trace form

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

$2q - 4q^{2} + 2q^{3} + 8q^{4} + 8q^{5} - 4q^{6} + 2q^{7} - 16q^{8} + 2q^{9} - 16q^{10} + 70q^{11} + 8q^{12} + 64q^{13} - 4q^{14} + 80q^{15} + 32q^{16} + 148q^{17} - 4q^{18} + 22q^{19} + 32q^{20} + 20q^{21} - 140q^{22} + 132q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} - 10x - 122$

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
$\operatorname{tr} a_p$	−4	2	8	2	70	64	148	22	132	−48	−100	−352	−82	−16	306	384	604	−164	834	510	−1092	−18	−144	−68	−1180

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.105 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, 12, 34] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(82, 4)[2]   # 82.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 82:4 1000