Atlas › Modular forms › Weight 2 › Level 524 › 524.2.a › 524.2.a.d

Newform orbit 524.2.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 3q^{3} + 2q^{5} + 7q^{7} + q^{9} + 12q^{11} - q^{13} - 10q^{15} + 10q^{17} + 11q^{21} - 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{17} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{4} - 5x^{3} - 6227x^{2} - 120381x + 1754127$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 524$ )
131	$+1$

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
$\operatorname{tr} a_p$	0	3	2	7	12	−1	10	0	−6	0	18	−2	−9	7	14	−20	−15	7	6	−16	14	28	−26	−4	12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 131, 262, 524]: [0, 1, 2, 12, 35, 69] (certified at ell = 2147482481)
T = [(3, 1), (5, 3), (17, 5), (19, 7), (23, 9)]; h of degree 11 = 1 x 11 (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(524, 2)[3]   # 524.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 524:2 1000