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

Newform orbit 63.6.a.d

$q$ -expansion

$q + 6q^{2} + 4q^{4} - 78q^{5} + 49q^{7} - 168q^{8} - 468q^{10} - 444q^{11} - 442q^{13} + 294q^{14} - 1136q^{16} + 126q^{17} + 2684q^{19} - 312q^{20} - 2664q^{22} - 4200q^{23} + O(q^{25})$

Coefficient field

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

$x + 72$

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
$\operatorname{tr} a_p$	6	0	−78	49	−444	−442	126	2684	−4200	5442	80	−5434	−7962	−11524	13920	9594	−27492	49478	−59356	−32040	−61846	−65776	−40188	7974	−143662

$\operatorname{tr} a_n$ 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)[3]   # 63.6.a.d

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

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