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

Newform orbit 126.4.a.b

$q$ -expansion

$q - 2q^{2} + 4q^{4} - 6q^{5} + 7q^{7} - 8q^{8} + 12q^{10} - 30q^{11} + 2q^{13} - 14q^{14} + 16q^{16} - 66q^{17} - 52q^{19} - 24q^{20} + 60q^{22} - 114q^{23} + O(q^{25})$

Coefficient field

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

$x + 96$

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	−6	7	−30	2	−66	−52	−114	−72	−196	−286	378	164	228	348	348	−106	596	−630	−1042	−88	1440	−1374	−34

$\operatorname{tr} a_n$ 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)[1]   # 126.4.a.b

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

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