Atlas › Modular forms › Weight 2 › Level 922 › 922.2.a › 922.2.a.e

Newform orbit 922.2.a.e

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{8} - 5x^{7} - 23x^{6} + 113x^{5} + 103x^{4} - 680x^{3} + 172x^{2} + 872x - 416$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
461	$-1$

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
$\operatorname{tr} a_p$	−8	2	3	3	9	−3	18	7	24	9	−12	−9	10	17	20	32	13	−13	11	15	−9	−7	39	4	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 461, 922]: [0, 1, 39, 117] (certified at ell = 2147478721)
T = [(3, 1), (5, 1)]; h of degree 38 = 1 x 38 (phi(ord eps) x dim S_k^new); 5 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(922, 2)[4]   # 922.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 922:2 1000