Atlas › Modular forms › Weight 2 › Level 589 › 589.2.a › 589.2.a.e

Newform orbit 589.2.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 7q^{3} + 18q^{4} + 9q^{5} + 9q^{6} + 6q^{7} + 21q^{9} - 2q^{10} + 18q^{11} + 4q^{12} - 3q^{13} + 7q^{14} + 22q^{16} + 8q^{17} - 28q^{18} - 14q^{19} + 24q^{20} - 12q^{21} - 8q^{22} + 19q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 7x^{13} - 39x^{12} + 371x^{11} + 131x^{10} - 6124x^{9} + 8206x^{8} + 31156x^{7} - 71630x^{6} - 10125x^{5} + 102319x^{4} - 45679x^{3} - 14328x^{2} + 6152x - 388$

Atkin–Lehner signs

$p$	$w_p$
19	$+1$
31	$-1$

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
$\operatorname{tr} a_p$	0	7	9	6	18	−3	8	−14	19	−4	14	−3	1	−18	37	−10	63	2	10	17	−16	−27	31	−20	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 19, 31, 589]: [0, 2, 3, 54] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(589, 2)[4]   # 589.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 589:2 1000