Atlas › Modular forms › Weight 4 › Level 20 › 20.4.a › 20.4.a.a

Newform orbit 20.4.a.a

$q$ -expansion

$q + 4q^{3} + 5q^{5} - 16q^{7} - 11q^{9} - 60q^{11} + 86q^{13} + 20q^{15} + 18q^{17} + 44q^{19} - 64q^{21} + 48q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{7}$ is a root of

$x + 44$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 20$ )
5	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	4	5	−16	−60	86	18	44	48	−186	176	254	186	−100	168	−498	−252	−58	−1036	168	506	272	948	−1014	−766

$\operatorname{tr} a_n$ 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 20, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.12 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, 4, 5, 10, 20]: [1, 2, 3, 3, 7, 12] (certified at ell = 2147483629)
T = [(3, 1), (7, 3)]; h of degree 1 = 1 x 1 (phi(ord eps) x dim S_k^new); 1 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(20, 4)[0]   # 20.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 20:4 1000