Atlas › Modular forms › Weight 4 › Level 63 › 63.4.a › 63.4.a.b

Newform orbit 63.4.a.b

$q$ -expansion

$q + q^{2} - 7q^{4} - 16q^{5} - 7q^{7} - 15q^{8} - 16q^{10} + 8q^{11} + 28q^{13} - 7q^{14} + 41q^{16} - 54q^{17} - 110q^{19} + 112q^{20} + 8q^{22} - 48q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{5}$ is a root of

$x + 47$

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	−16	−7	8	28	−54	−110	−48	110	12	−246	−182	128	−324	162	−810	−488	244	768	−702	440	1302	−730	294

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{3/2}$ 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 0.107 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, 2, 3, 4, 10, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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, 4)[1]   # 63.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 63:4 1000