Atlas › Modular forms › Weight 4 › Level 45 › 45.4.a › 45.4.a.d

Newform orbit 45.4.a.d

$q$ -expansion

$q + 4q^{2} + 8q^{4} + 5q^{5} + 6q^{7} + 20q^{10} - 32q^{11} - 38q^{13} + 24q^{14} - 64q^{16} - 26q^{17} + 100q^{19} + 40q^{20} - 128q^{22} + 78q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{7}$ is a root of

$x - 22$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 45$ )
5	$-1$

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
$\operatorname{tr} a_p$	4	0	5	6	−32	−38	−26	100	78	50	−108	266	−22	442	514	−2	−500	−518	126	−412	−878	600	−282	150	386

$\operatorname{tr} a_n$ 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 45, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 5, 9, 15, 45]: [1, 2, 3, 4, 8, 20] (certified at ell = 2147483629)
T = [(2, 1), (7, 3)]; h of degree 5 = 1 x 5 (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(45, 4)[3]   # 45.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 45:4 1000