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

Newform orbit 509.2.a.a

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 3q^{2} - 5q^{3} + 3q^{4} - 5q^{5} - 8q^{6} - 17q^{7} - 6q^{8} + 5q^{9} - 7q^{10} - 20q^{11} - 3q^{12} - 6q^{13} + 4q^{14} - 11q^{15} - 15q^{16} + q^{17} - 10q^{18} - 45q^{19} - 6q^{20} - 15q^{21} - 4q^{23} - 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 8x^{13} + x^{12} - 109x^{11} - 117x^{10} + 508x^{9} + 504x^{8} - 1193x^{7} - 621x^{6} + 1445x^{5} + 12x^{4} - 682x^{3} + 232x^{2} + 21x - 13$

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 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
$\operatorname{tr} a_p$	−3	−5	−5	−17	−20	−6	1	−45	−4	−8	−17	−8	6	−45	4	3	−11	−11	−47	−15	7	−37	7	11	14

$\operatorname{tr} a_n$ 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)[0]   # 509.2.a.a

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

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