Atlas › Modular forms › Weight 2 › Level 571 › 571.2.a › 571.2.a.h

Newform orbit 571.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{10} + 116x^{9} + 4283x^{8} + 21851x^{7} - 1892153x^{6} - 30503241x^{5} + 124599657x^{4} + 4183470151x^{3} + 10844738910x^{2} - 12783182636x - 41511368420$

Atkin–Lehner signs

$p$	$w_p$
571	$+1$

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
$\operatorname{tr} a_p$	−1	−5	−13	−5	−3	−2	1	−3	−16	−34	1	−12	−16	19	−15	−38	−22	12	14	5	−9	−65	0	−27	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 0.312 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, 571]: [0, 48] (certified at ell = 2147477851)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 10 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(571, 2)[7]   # 571.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 571:2 1000