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

Newform orbit 82.4.a.d

Trace form

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

$3q - 6q^{2} - 4q^{3} + 12q^{4} - 2q^{5} + 8q^{6} - 26q^{7} - 24q^{8} + 11q^{9} + 4q^{10} - 34q^{11} - 16q^{12} - 130q^{13} + 52q^{14} - 270q^{15} + 48q^{16} - 214q^{17} - 22q^{18} - 28q^{19} - 8q^{20} + 24q^{21} + 68q^{22} - 216q^{23} + 32q^{24} + O(q^{25})$

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

$x^{3} + 6x^{2} - 200x - 1050$

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
$\operatorname{tr} a_p$	−6	−4	−2	−26	−34	−130	−214	−28	−216	−294	308	50	123	288	−408	−74	−236	786	290	−808	−198	−440	−296	2130	−762

$\operatorname{tr} a_n$ 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)[3]   # 82.4.a.d

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

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