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

Newform orbit 571.2.a.i

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 4q^{3} + 22q^{4} + 26q^{5} - 6q^{6} - 4q^{7} + 6q^{8} + 34q^{9} + 7q^{10} - 6q^{11} - 12q^{12} + 6q^{13} + 23q^{14} + 10q^{15} + 30q^{16} + 20q^{17} - 24q^{18} - 15q^{19} + 51q^{20} + 20q^{21} - 4q^{22} + q^{23} - 29q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{18} - 114x^{17} - 282x^{16} + 445195x^{15} - 10656264x^{14} - 565499038x^{13} + 23232216942x^{12} + 174516573544x^{11} - 18576625693213x^{10} + 149155995753815x^{9} + 5352702534744041x^{8} - 103152844806741442x^{7} + 31197361895123547x^{6} + 13460034295649199593x^{5} - 110277016448480109745x^{4} + 10075032992020957949x^{3} + 3265027640220622988591x^{2} - 12215237689437400357901x + 9714647710614367915166$

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
$\operatorname{tr} a_p$	0	4	26	−4	−6	6	20	−15	1	21	14	29	66	−14	3	39	4	−8	7	14	0	−14	4	15	28

$\operatorname{tr} a_n$ 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)[8]   # 571.2.a.i

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

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