Atlas › Modular forms › Weight 2 › Level 511 › 511.2.a › 511.2.a.e

Newform orbit 511.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - x^{8} - 38x^{7} + 37x^{6} + 393x^{5} - 143x^{4} - 1485x^{3} - 232x^{2} + 1408x + 277$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
73	$+1$

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	−1	7	9	−5	−1	14	0	−4	31	9	−3	59	−19	26	1	25	22	−24	−20	−9	−22	23	25	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.2 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, 7, 73, 511]: [0, 1, 6, 50] (certified at ell = 2147482873)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 4 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(511, 2)[4]   # 511.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 511:2 1000