Atlas › Modular forms › Weight 6 › Level 63 › 63.6.a › 63.6.a.b

Newform orbit 63.6.a.b

$q$ -expansion

$q - 5q^{2} - 7q^{4} - 94q^{5} - 49q^{7} + 195q^{8} + 470q^{10} - 52q^{11} - 770q^{13} + 245q^{14} - 751q^{16} + 2022q^{17} + 1732q^{19} + 658q^{20} + 260q^{22} + 576q^{23} + O(q^{25})$

Coefficient field

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

$x + 99$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 63$ )
7	$+1$

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
$\operatorname{tr} a_p$	−5	0	−94	−49	−52	−770	2022	1732	576	−5518	6336	−7338	3262	5420	−864	−4182	11220	−45602	1396	−18720	46362	97424	81228	3182	4914

$\operatorname{tr} a_n$ 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 63, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 7, 9, 21, 63]: [1, 3, 5, 6, 16, 42] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 13 = 1 x 13 (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(63, 6)[1]   # 63.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 63:6 1000