Atlas › Modular forms › Weight 2 › Level 581 › 581.2.a › 581.2.a.b

Newform orbit 581.2.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 7q^{3} + 2q^{4} - 5q^{5} - 4q^{6} + 7q^{7} + 6q^{9} - 12q^{10} - 9q^{11} - 10q^{12} - 18q^{13} + 9q^{15} + 4q^{16} - 3q^{17} + 7q^{18} - 29q^{19} + 9q^{20} - 7q^{21} - q^{22} + q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 7x^{6} + 7x^{5} - 30x^{4} - 36x^{3} + 39x^{2} + 24x - 13$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
83	$-1$

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
$\operatorname{tr} a_p$	0	−7	−5	7	−9	−18	−3	−29	1	−6	−21	−5	6	−11	16	11	−10	−21	−13	12	−2	−7	7	12	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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, 7, 83, 581]: [0, 1, 8, 58] (certified at ell = 2147483323)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 4 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(581, 2)[1]   # 581.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 581:2 1000