Atlas › Modular forms › Weight 2 › Level 459 › 459.2.a › 459.2.a.n

Newform orbit 459.2.a.n

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + q^{2} + 9q^{4} + 3q^{5} - 4q^{7} - 9q^{8} + 11q^{10} + 4q^{11} + q^{13} + 18q^{14} + 13q^{16} - 3q^{17} + 5q^{19} - 7q^{20} - 8q^{22} + 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{3} - 43x^{2} - 249x + 15669$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 459$ )
17	$+1$

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
$\operatorname{tr} a_p$	1	0	3	−4	4	1	−3	5	3	−3	0	12	−3	5	−4	4	−30	36	−29	−19	0	14	4	30	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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, 3, 9, 17, 27, 51, 153, 459]: [0, 1, 2, 2, 4, 8, 20, 56] (certified at ell = 2147482801)
T = [(2, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (23, 7), (29, 8)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 6 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(459, 2)[13]   # 459.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 459:2 1000