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

Newform orbit 571.2.a.g

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - q^{2} - 3q^{3} + 9q^{4} - 9q^{5} - 8q^{6} - 3q^{7} + 6q^{8} + 5q^{9} - 8q^{10} - 12q^{11} - 25q^{12} - 5q^{13} - q^{14} + 3q^{15} + 7q^{16} - 13q^{17} + 24q^{18} + 8q^{19} - 29q^{20} - 5q^{21} - 17q^{22} - q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{6} + 76x^{5} + 1574x^{4} + 4352x^{3} - 135371x^{2} - 625636x + 2640511$

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	−3	−9	−3	−12	−5	−13	8	−1	−10	−20	2	−34	−14	8	−1	15	−31	21	−49	5	51	−23	−4	15

$\operatorname{tr} a_n$ 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)[6]   # 571.2.a.g

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

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