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

Newform orbit 587.2.a.c

Trace form

The 31 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$31q + 6q^{2} + 2q^{3} + 38q^{4} + 7q^{5} - 4q^{6} + 19q^{7} + 15q^{8} + 35q^{9} + 6q^{10} + 2q^{11} - 5q^{12} + 51q^{13} + q^{15} + 48q^{16} + 13q^{17} + 15q^{18} + 15q^{19} - 3q^{20} + 26q^{21} + 26q^{22} + q^{23} - 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 31. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{31} - 8x^{30} - 78x^{29} + 747x^{28} + 2226x^{27} - 29097x^{26} - 24831x^{25} + 628767x^{24} - 73063x^{23} - 8443758x^{22} + 5227833x^{21} + 74892263x^{20} - 66394187x^{19} - 455561581x^{18} + 465908863x^{17} + 1944329198x^{16} - 2107023535x^{15} - 5891763932x^{14} + 6485639578x^{13} + 12705899910x^{12} - 13855675179x^{11} - 19358283998x^{10} + 20501622315x^{9} + 20443566763x^{8} - 20536815541x^{7} - 14443941272x^{6} + 13222192173x^{5} + 6418161242x^{4} - 4903545446x^{3} - 1602180139x^{2} + 788541831x + 173143604$

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 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
$\operatorname{tr} a_p$	6	2	7	19	2	51	13	15	1	14	9	85	−5	26	7	36	−13	24	66	−11	19	23	1	−14	56

$\operatorname{tr} a_n$ 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)[2]   # 587.2.a.c

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

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