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

Newform orbit 922.2.a.f

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{14} - 9x^{13} - 54x^{12} + 748x^{11} - 454x^{10} - 18368x^{9} + 61570x^{8} + 66532x^{7} - 743926x^{6} + 1642036x^{5} - 1529445x^{4} + 335197x^{3} + 434368x^{2} - 300292x + 52072$

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
$\operatorname{tr} a_p$	14	3	6	7	6	4	13	−18	26	9	8	14	−4	−2	−3	23	−21	−8	−26	9	−9	−14	28	0	−19

$\operatorname{tr} a_n$ 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)[5]   # 922.2.a.f

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

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