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

Newform orbit 82.6.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 12q^{2} - 28q^{3} + 48q^{4} - 26q^{5} - 112q^{6} - 158q^{7} + 192q^{8} - 61q^{9} - 104q^{10} - 846q^{11} - 448q^{12} - 1258q^{13} - 632q^{14} - 682q^{15} + 768q^{16} - 2166q^{17} - 244q^{18} - 932q^{19} - 416q^{20} - 2776q^{21} - 3384q^{22} - 1960q^{23} - 1792q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 54x^{2} + 612x + 1018$

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 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
$\operatorname{tr} a_p$	12	−28	−26	−158	−846	−1258	−2166	−932	−1960	−3294	4764	970	5043	1520	−9544	−10178	5756	25514	25342	59272	95626	15448	88536	−88446	−323962

$\operatorname{tr} a_n$ 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)[0]   # 82.6.a.a

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

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