Atlas › Modular forms › Weight 4 › Level 225 › 225.4.a › 225.4.a.f

Newform orbit 225.4.a.f

$q$ -expansion

$q + q^{2} - 7q^{4} + 24q^{7} - 15q^{8} - 52q^{11} - 22q^{13} + 24q^{14} + 41q^{16} - 14q^{17} - 20q^{19} - 52q^{22} - 168q^{23} + O(q^{25})$

Coefficient field

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

$x - 73$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 225$ )
5	— ( $5^2 \mid 225$ )

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	0	24	−52	−22	−14	−20	−168	−230	−288	34	−122	188	256	−338	−100	742	84	328	38	−240	1212	−330	−866

$\operatorname{tr} a_n$ 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 225, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 25, 45, 75, 225]: [1, 2, 3, 4, 8, 9, 20, 32, 92] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 5 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(225, 4)[5]   # 225.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 225:4 1000