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

Newform orbit 63.6.a.c

$q$ -expansion

$q - q^{2} - 31q^{4} + 34q^{5} - 49q^{7} + 63q^{8} - 34q^{10} + 340q^{11} + 454q^{13} + 49q^{14} + 929q^{16} + 798q^{17} + 892q^{19} - 1054q^{20} - 340q^{22} + 3192q^{23} + O(q^{25})$

Coefficient field

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

$x - 33$

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
$\operatorname{tr} a_p$	−1	0	34	−49	340	454	798	892	3192	8242	−2496	9798	−19834	−17236	−8928	−150	42396	14758	−1676	−14568	78378	−2272	37764	117286	10002

$\operatorname{tr} a_n$ 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)[2]   # 63.6.a.c

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

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