Atlas › Modular forms › Weight 6 › Level 99 › 99.6.a › 99.6.a.c

Newform orbit 99.6.a.c

$q$ -expansion

$q + 4q^{2} - 16q^{4} + 19q^{5} + 10q^{7} - 192q^{8} + 76q^{10} + 121q^{11} - 1148q^{13} + 40q^{14} - 256q^{16} - 686q^{17} - 384q^{19} - 304q^{20} + 484q^{22} - 3709q^{23} + O(q^{25})$

Coefficient field

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

$x - 23$

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
$\operatorname{tr} a_p$	4	0	19	10	121	−1148	−686	−384	−3709	5424	−6443	12063	1528	−4026	−7168	29862	6461	−16980	29999	−31023	1924	65138	102714	−17415	66905

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.283 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, 3, 6, 6, 22, 62] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 5 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, 6)[2]   # 99.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 99:6 1000