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

Newform orbit 571.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{4} - 14x^{3} - 640x^{2} + 3592x + 91860$

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 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
$\operatorname{tr} a_p$	2	2	4	−2	8	12	−8	4	4	24	0	−20	8	−12	−20	26	16	36	−10	−16	26	−12	−8	26	−12

$\operatorname{tr} a_n$ 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)[5]   # 571.2.a.f

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

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