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

Newform orbit 583.2.a.i

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + q^{2} + 6q^{3} + 15q^{4} + 10q^{5} - 5q^{6} - 3q^{7} + 3q^{8} + 18q^{9} + 4q^{10} + 12q^{11} + 9q^{12} - 2q^{13} - 16q^{14} + 19q^{15} + 25q^{16} + 28q^{17} - 28q^{18} + 4q^{19} + 47q^{20} - 10q^{21} + q^{22} + 5q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{12} - 19x^{11} - 67x^{10} + 3287x^{9} - 12767x^{8} - 131783x^{7} + 958409x^{6} + 668419x^{5} - 18838176x^{4} + 32891720x^{3} + 73559312x^{2} - 233399840x + 129189216$

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
$\operatorname{tr} a_p$	1	6	10	−3	12	−2	28	4	5	7	−4	−2	11	−6	49	−12	−10	−30	8	−12	33	−6	14	33	57

$\operatorname{tr} a_n$ 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)[8]   # 583.2.a.i

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

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