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

Newform orbit 82.6.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 20q^{2} - 10q^{3} + 80q^{4} - 38q^{5} + 40q^{6} - 38q^{7} - 320q^{8} + 265q^{9} + 152q^{10} + 416q^{11} - 160q^{12} + 268q^{13} + 152q^{14} + 22q^{15} + 1280q^{16} - 2198q^{17} - 1060q^{18} - 2250q^{19} - 608q^{20} - 6344q^{21} - 1664q^{22} - 12008q^{23} + 640q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 48x^{4} - 2492x^{3} - 117954x^{2} - 737450x + 8103036$

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
$\operatorname{tr} a_p$	−20	−10	−38	−38	416	268	−2198	−2250	−12008	−11684	−6780	−5822	−8405	13248	−2000	−32036	−41376	45314	47244	−87348	44550	−6864	−119888	−136626	7086

$\operatorname{tr} a_n$ 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)[2]   # 82.6.a.c

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

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