Atlas › Modular forms › Weight 2 › Level 592 › 592.2.a › 592.2.a.j

Newform orbit 592.2.a.j

Trace form

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

$4q - 2q^{3} + 5q^{5} + q^{7} + 8q^{9} - 4q^{11} + 5q^{13} + 8q^{17} - 2q^{19} + q^{21} + 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ , whose minimal polynomial is

$x^{4} - 35x^{3} - 2654x^{2} + 23132x + 277664$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 592$ )
37	$-1$

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	−2	5	1	−4	5	8	−2	9	7	1	4	2	−6	29	−5	10	−1	1	17	8	−15	−15	−20	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.381 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, 8, 16, 37, 74, 148, 296, 592]: [0, 1, 2, 3, 4, 3, 11, 22, 42, 80] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 18 = 1 x 18 (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(592, 2)[9]   # 592.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 592:2 1000