Atlas › Modular forms › Weight 6 › Level 98 › 98.6.a › 98.6.a.a

Newform orbit 98.6.a.a

$q$ -expansion

$q - 4q^{2} - 10q^{3} + 16q^{4} - 84q^{5} + 40q^{6} - 64q^{8} - 143q^{9} + 336q^{10} - 336q^{11} - 160q^{12} - 584q^{13} + 840q^{15} + 256q^{16} + 1458q^{17} + 572q^{18} - 470q^{19} - 1344q^{20} + 1344q^{22} - 4200q^{23} + 640q^{24} + O(q^{25})$

Coefficient field

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

$x + 94$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	— ( $7^2 \mid 98$ )

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	−10	−84	0	−336	−584	1458	−470	−4200	4866	7372	14330	−6222	3704	1812	−37242	−34302	−24476	−17452	28224	−3602	42872	35202	−26730	16978

$\operatorname{tr} a_n$ 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 98, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 7, 14, 49, 98]: [1, 2, 5, 12, 25, 72] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (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(98, 6)[0]   # 98.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 98:6 1000