Atlas › Modular forms › Weight 2 › Level 509 › 509.2.a › 509.2.a.b

Newform orbit 509.2.a.b

Trace form

The 28 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$28q + 3q^{2} + 5q^{3} + 41q^{4} + q^{5} + 2q^{6} + 19q^{7} + 6q^{8} + 37q^{9} + q^{10} + 24q^{11} - 3q^{12} + 8q^{13} - 2q^{14} + 5q^{15} + 59q^{16} - q^{17} - 4q^{18} + 61q^{19} - 12q^{20} + 5q^{21} + 10q^{22} - 4q^{23} + q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 28. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{28} - 8x^{27} - 79x^{26} + 775x^{25} + 2020x^{24} - 29956x^{23} - 9270x^{22} + 607348x^{21} - 451466x^{20} - 7233548x^{19} + 9489672x^{18} + 53890768x^{17} - 89144249x^{16} - 258670457x^{15} + 486611251x^{14} + 798783953x^{13} - 1647657054x^{12} - 1521661510x^{11} + 3479175501x^{10} + 1574898974x^{9} - 4402183614x^{8} - 508093292x^{7} + 2997640229x^{6} - 394588162x^{5} - 840597920x^{4} + 243169571x^{3} + 43794823x^{2} - 17566912x + 839924$

Atkin–Lehner signs

$p$	$w_p$
509	$-1$

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
$\operatorname{tr} a_p$	3	5	1	19	24	8	−1	61	−4	−6	19	6	2	59	−8	−11	15	19	81	9	5	63	19	−23	−4

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 509]: [0, 43] (certified at ell = 2147468909)
T = [(2, 1), (3, 1)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 5 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(509, 2)[1]   # 509.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 509:2 1000