Atlas › Modular forms › Weight 2 › Level 587 › 587.2.a › 587.2.a.b

Newform orbit 587.2.a.b

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 3q^{2} - 2q^{3} + 5q^{4} - 11q^{5} - 8q^{6} - 15q^{7} - 6q^{8} + 9q^{9} - 10q^{10} - 5q^{11} + 5q^{12} - 21q^{13} + 11q^{14} - 3q^{15} + 5q^{16} - 13q^{17} - 17q^{18} - 18q^{19} - 17q^{20} - 31q^{21} - 23q^{22} - 3q^{23} - 34q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} + 5x^{12} - 19x^{11} - 100x^{10} + 128x^{9} + 646x^{8} - 398x^{7} - 1551x^{6} + 761x^{5} + 1251x^{4} - 695x^{3} + 18x^{2} + 19x + 1$

Atkin–Lehner signs

$p$	$w_p$
587	$+1$

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 odd 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
$\operatorname{tr} a_p$	−3	−2	−11	−15	−5	−21	−13	−18	−3	2	10	−69	13	−47	13	−59	17	−30	−4	7	11	22	−12	30	−65

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 587]: [0, 50] (certified at ell = 2147470837)
T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (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(587, 2)[1]   # 587.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 587:2 1000