Atlas › Modular forms › Weight 4 › Level 156 › 156.4.a › 156.4.a.a

Newform orbit 156.4.a.a

$q$ -expansion

$q - 3q^{3} - 6q^{5} - 4q^{7} + 9q^{9} + 36q^{11} + 13q^{13} + 18q^{15} + 66q^{17} + 56q^{19} + 12q^{21} + 96q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x + 18$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 156$ )
3	$+1$
13	$-1$

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	−3	−6	−4	36	13	66	56	96	222	260	−106	−90	44	168	30	348	−346	−256	−168	−814	200	1236	318	−502

$\operatorname{tr} a_n$ 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 156, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.154 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, 4, 6, 12, 13, 26, 39, 52, 78, 156]: [1, 2, 2, 3, 5, 9, 5, 13, 16, 24, 46, 90] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 6 = 1 x 6 (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(156, 4)[0]   # 156.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 156:4 1000