Atlas › Modular forms › Weight 6 › Level 100 › 100.6.a › 100.6.a.a

Newform orbit 100.6.a.a

$q$ -expansion

$q - 22q^{3} - 218q^{7} + 241q^{9} - 480q^{11} + 622q^{13} - 186q^{17} - 1204q^{19} + 4796q^{21} + 3186q^{23} + O(q^{25})$

Coefficient field

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

$x + 240$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 100$ )
5	— ( $5^2 \mid 100$ )

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	−22	0	−218	−480	622	−186	−1204	3186	5526	9356	−5618	−14394	370	−16146	4374	−11748	13202	11542	−29532	−33698	31208	38466	119514	−94658

$\operatorname{tr} a_n$ 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 100, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.29 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, 4, 5, 10, 20, 25, 50, 100]: [1, 2, 4, 3, 9, 18, 13, 39, 78] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; 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(100, 6)[0]   # 100.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 100:6 1000