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

Newform orbit 581.2.a.c

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 7q^{3} + 16q^{4} + 5q^{5} + 13q^{7} - 6q^{8} + 14q^{9} + 6q^{10} + 3q^{11} + 4q^{12} + 20q^{13} + q^{15} + 22q^{16} - 3q^{17} - 17q^{18} + 29q^{19} + 5q^{20} + 7q^{21} + 3q^{22} - 7q^{23} + 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 7x^{12} - 23x^{11} + 256x^{10} - 207x^{9} - 2060x^{8} + 3752x^{7} + 4890x^{6} - 14045x^{5} - 33x^{4} + 14708x^{3} - 3586x^{2} - 3644x - 413$

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 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	5	13	3	20	−3	29	−7	4	25	−19	6	21	12	−11	12	23	21	4	30	29	−13	0	19

$\operatorname{tr} a_n$ 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)[2]   # 581.2.a.c

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

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