Atlas › Modular forms › Weight 2 › Level 583 › 583.2.a › 583.2.a.h

Newform orbit 583.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{10} - 211x^{8} - 102x^{7} + 15853x^{6} + 11230x^{5} - 509471x^{4} - 334360x^{3} + 6695972x^{2} + 2426144x - 29269232$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
53	$-1$

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
$\operatorname{tr} a_p$	3	−1	8	3	−10	3	31	11	10	4	2	−1	43	12	3	10	−4	26	−26	7	33	−7	55	33	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 0.221 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, 11, 53, 583]: [0, 2, 5, 56] (certified at ell = 2147482481)
T = [(2, 1), (3, 3)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 6 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(583, 2)[7]   # 583.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 583:2 1000