Atlas › Modular forms › Weight 2 › Level 586 › 586.2.a › 586.2.a.g

Newform orbit 586.2.a.g

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 10q^{2} - q^{3} + 10q^{4} + 3q^{5} - q^{6} + 2q^{7} + 10q^{8} + 19q^{9} + 3q^{10} + 7q^{11} - q^{12} + 9q^{13} + 2q^{14} + 4q^{15} + 10q^{16} + 8q^{17} + 19q^{18} - 7q^{19} + 3q^{20} + 8q^{21} + 7q^{22} + 10q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{10} - 2x^{9} - 60x^{8} + 208x^{7} + 708x^{6} - 4170x^{5} + 4959x^{4} + 3110x^{3} - 9647x^{2} + 6292x - 1331$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
293	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	10	−1	3	2	7	9	8	−7	10	27	2	13	6	−19	−6	15	−5	−19	−45	4	0	8	−21	14	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 0.212 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, 293, 586]: [0, 1, 25, 75] (certified at ell = 2147478961)
T = [(3, 1), (5, 1)]; h of degree 24 = 1 x 24 (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(586, 2)[6]   # 586.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 586:2 1000