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

Newform orbit 511.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 2x^{5} - 8x^{4} - 10x^{3} + 13x^{2} + 2x - 1$

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 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	1	−2	−6	−9	−4	−13	0	−3	−35	−2	6	−27	−7	11	−17	−6	6	−15	−13	−6	15	20	−15	23

$\operatorname{tr} a_n$ 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)[3]   # 511.2.a.d

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

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