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

Newform orbit 82.4.a.b

$q$ -expansion

$q + 2q^{2} + 10q^{3} + 4q^{4} - 6q^{5} + 20q^{6} - 10q^{7} + 8q^{8} + 73q^{9} - 12q^{10} - 54q^{11} + 40q^{12} - 82q^{13} - 20q^{14} - 60q^{15} + 16q^{16} + 42q^{17} + 146q^{18} + 134q^{19} - 24q^{20} - 100q^{21} - 108q^{22} + 48q^{23} + 80q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x - 4$

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
$\operatorname{tr} a_p$	2	10	−6	−10	−54	−82	42	134	48	30	−136	2	41	200	−30	390	−444	38	−610	−42	110	950	900	138	170

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 82, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[1]   # 82.4.a.b

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

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