Atlas › Modular forms › Weight 2 › Level 484 › 484.2.a › 484.2.a.d

Newform orbit 484.2.a.d

Trace form

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

$2q + q^{3} + 3q^{5} + 11q^{9} - 15q^{15} + 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{2} - 10x - 8$

Complex multiplication by $\mathbb{Q}(\sqrt{-11})$ : $a_p = 0$ 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 83 inert primes $p < 1000$ . This is evidence from the stored coefficients, not a proof.

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 484$ )
11	— ( $11^2 \mid 484$ )

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	1	3	0	0	0	0	0	9	0	5	−7	0	0	−24	12	15	0	−13	3	0	0	0	9	−17

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 11, 22, 44, 121, 242, 484]: [0, 1, 2, 2, 5, 9, 12, 35, 69] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 9 = 1 x 9 (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(484, 2)[3]   # 484.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 484:2 1000