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

Newform orbit 225.4.a.d

$q$ -expansion

$q - 8q^{4} - 20q^{7} + 70q^{13} + 64q^{16} + 56q^{19} + O(q^{25})$

Coefficient field

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

$x + 60$

Complex multiplication by $\mathbb{Q}(\sqrt{-3})$ : $a_p = 0$ for every conjugate at the 5 primes $p \le 31$ inert in it (exactly, from $\operatorname{tr} a_{p^2} = -\dim \cdot p^{k-1}$ ), and $\operatorname{tr} a_p = 0$ at all 86 inert primes $p < 1000$ . This is evidence from the stored coefficients, not a proof.

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
$\operatorname{tr} a_p$	0	0	0	−20	0	70	0	56	0	0	308	−110	0	520	0	0	0	182	880	0	−1190	884	0	0	1330

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

Sato–Tate

With CM the normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 225, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ (CM forms follow a different law):

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)[3]   # 225.4.a.d

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

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