Atlas › Modular forms › Weight 6 › Level 4 › 4.6.a › 4.6.a.a

Newform orbit 4.6.a.a

$q$ -expansion

$q - 12q^{3} + 54q^{5} - 88q^{7} - 99q^{9} + 540q^{11} - 418q^{13} - 648q^{15} + 594q^{17} + 836q^{19} + 1056q^{21} - 4104q^{23} + O(q^{25})$

Coefficient field

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

$x - 42$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 4$ )

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	−12	54	−88	540	−418	594	836	−4104	−594	4256	−298	17226	−12100	−1296	19494	−7668	−34738	21812	−46872	67562	−76912	67716	29754	−122398

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 4, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.095 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]: [1, 2, 4] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; 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(4, 6)[0]   # 4.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 4:6 1000