Atlas › Modular forms › Weight 4 › Level 99 › 99.4.a › 99.4.a.a

Newform orbit 99.4.a.a

$q$ -expansion

$q + q^{2} - 7q^{4} + 4q^{5} - 26q^{7} - 15q^{8} + 4q^{10} - 11q^{11} - 32q^{13} - 26q^{14} + 41q^{16} - 74q^{17} - 60q^{19} - 28q^{20} - 11q^{22} + 182q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{5}$ is a root of

$x - 13$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 99$ )
11	$+1$

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
$\operatorname{tr} a_p$	1	0	4	−26	−11	−32	−74	−60	182	90	−8	−66	−422	408	506	−348	200	132	−1036	−762	−542	−550	132	−570	14

$\operatorname{tr} a_n$ 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 99, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 3, 9, 11, 33, 99]: [1, 2, 4, 4, 14, 38] (certified at ell = 2147482951)
T = [(2, 1), (5, 3)]; h of degree 12 = 1 x 12 (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(99, 4)[0]   # 99.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 99:4 1000