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

Newform orbit 82.6.a.b

Trace form

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

$4q - 16q^{2} + 8q^{3} + 64q^{4} + 12q^{5} - 32q^{6} + 158q^{7} - 256q^{8} + 184q^{9} - 48q^{10} + 2q^{11} + 128q^{12} - 900q^{13} - 632q^{14} + 762q^{15} + 1024q^{16} + 1660q^{17} - 736q^{18} - 1792q^{19} + 192q^{20} + 7184q^{21} - 8q^{22} + 10264q^{23} - 512q^{24} + O(q^{25})$

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

$x^{4} - 20x^{3} - 11792x^{2} + 217486x + 5613840$

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
$\operatorname{tr} a_p$	−16	8	12	158	2	−900	1660	−1792	10264	13080	7140	−4624	6724	−9828	−8048	21676	−52168	−118416	−12658	1720	−24484	52304	63124	−10644	−183192

$\operatorname{tr} a_n$ 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)[1]   # 82.6.a.b

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

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