Atlas › Modular forms › Weight 4 › Level 126 › 126.4.a › 126.4.a.f

Newform orbit 126.4.a.f

$q$ -expansion

$q + 2q^{2} + 4q^{4} - 22q^{5} - 7q^{7} + 8q^{8} - 44q^{10} - 26q^{11} - 54q^{13} - 14q^{14} + 16q^{16} - 74q^{17} + 116q^{19} - 88q^{20} - 52q^{22} + 58q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{11}$ is a root of

$x + 100$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 126$ )
7	$+1$

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
$\operatorname{tr} a_p$	2	0	−22	−7	−26	−54	−74	116	58	−208	−252	50	−126	164	444	−12	−124	−162	−860	238	−146	−984	−656	954	526

$\operatorname{tr} a_n$ 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 126, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.152 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, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126]: [1, 2, 2, 5, 3, 4, 8, 11, 10, 28, 26, 76] (certified at ell = 2147483647)
T = [(5, 1), (11, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(126, 4)[5]   # 126.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 126:4 1000