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

Newform orbit 225.4.a.n

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 2q^{2} + 24q^{4} + 26q^{7} + 84q^{8} - 28q^{11} + 18q^{13} - 126q^{14} + 120q^{16} + 68q^{17} + 6q^{19} + 124q^{22} - 132q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{19})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 80x - 699$

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
$\operatorname{tr} a_p$	2	0	0	26	−28	18	68	6	−132	−92	122	−284	−392	690	620	848	−124	750	−358	−824	−108	−880	−156	864	1042

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

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)[13]   # 225.4.a.n

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

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