Atlas › Modular forms › Weight 2 › Level 527 › 527.2.a › 527.2.a.e

Newform orbit 527.2.a.e

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 3q^{2} + 3q^{3} + 21q^{4} + 4q^{5} + 2q^{6} + 8q^{7} + 3q^{8} + 29q^{9} + q^{10} - 6q^{11} + 6q^{12} + 19q^{13} + 3q^{14} - 7q^{15} + 31q^{16} + 16q^{17} - 14q^{18} + 13q^{19} + 10q^{20} + 11q^{21} - 4q^{22} - 12q^{23} - 26q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 6x^{15} - 49x^{14} + 345x^{13} + 699x^{12} - 6845x^{11} - 2997x^{10} + 65227x^{9} - 12270x^{8} - 323313x^{7} + 154161x^{6} + 795532x^{5} - 507943x^{4} - 769812x^{3} + 581190x^{2} + 35441x - 19863$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$
31	$+1$

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
$\operatorname{tr} a_p$	3	3	4	8	−6	19	16	13	−12	−22	−16	18	−12	−13	24	4	3	10	42	−11	52	−21	9	19	85

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 17, 31, 527]: [0, 2, 3, 50] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (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(527, 2)[4]   # 527.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 527:2 1000