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

Newform orbit 511.2.a.f

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 4q^{2} - q^{3} + 14q^{4} + q^{5} + 2q^{6} - 10q^{7} + 6q^{8} + 39q^{9} - 6q^{10} + 5q^{11} + q^{12} - q^{13} - 4q^{14} + q^{15} + 14q^{16} + 8q^{17} + 26q^{18} + 2q^{19} - 16q^{20} + q^{21} - 3q^{22} + 10q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 3x^{9} - 49x^{8} + 118x^{7} + 798x^{6} - 1259x^{5} - 5370x^{4} + 4227x^{3} + 13282x^{2} - 3900x - 7551$

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
$\operatorname{tr} a_p$	4	−1	1	−10	5	−1	8	2	10	9	−17	−9	39	−7	−38	9	5	10	−22	30	10	0	−25	37	5

$\operatorname{tr} a_n$ 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)[5]   # 511.2.a.f

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

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