Atlas › Modular forms › Weight 2 › Level 515 › 515.2.a › 515.2.a.b

Newform orbit 515.2.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} + 11x^{7} - 84x^{6} - 898x^{5} + 2725x^{4} + 19830x^{3} - 45660x^{2} - 67401x + 103797$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
103	$+1$

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
$\operatorname{tr} a_p$	−2	−3	−8	−11	5	−23	−13	−12	−9	7	−4	−27	5	−2	5	−15	3	−42	0	23	−31	−14	−22	32	−40

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.205 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, 5, 103, 515]: [0, 1, 9, 54] (certified at ell = 2147482501)
T = [(2, 1), (3, 3)]; h of degree 35 = 1 x 35 (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(515, 2)[1]   # 515.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 515:2 1000