Atlas › Modular forms › Weight 2 › Level 866 › 866.2.a › 866.2.a.e

Newform orbit 866.2.a.e

Trace form

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

$16q + 16q^{2} + 4q^{3} + 16q^{4} + 6q^{5} + 4q^{6} + 3q^{7} + 16q^{8} + 30q^{9} + 6q^{10} + 6q^{11} + 4q^{12} + 13q^{13} + 3q^{14} + 2q^{15} + 16q^{16} + 13q^{17} + 30q^{18} + 14q^{19} + 6q^{20} + 10q^{21} + 6q^{22} - 14q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{16} - 79x^{15} - 4150x^{14} + 517974x^{13} - 6139105x^{12} - 714493189x^{11} + 24238403898x^{10} + 15555672356x^{9} - 11011081891072x^{8} + 118014235621660x^{7} + 1370174002643976x^{6} - 28343344544714512x^{5} + 25347976242819696x^{4} + 1897166162055925088x^{3} - 8586329708242123776x^{2} - 25637543316553961888x + 164317941233419690240$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
433	$+1$

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
$\operatorname{tr} a_p$	16	4	6	3	6	13	13	14	−14	14	15	19	11	5	−23	17	14	−4	15	−7	−10	−14	−13	−12	−22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.422 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, 2, 433, 866]: [0, 1, 36, 110] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 8 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(866, 2)[4]   # 866.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 866:2 1000