Atlas › Modular forms › Weight 4 › Level 114 › 114.4.a › 114.4.a.f

Newform orbit 114.4.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} + 6q^{3} + 8q^{4} + 11q^{5} + 12q^{6} + 9q^{7} + 16q^{8} + 18q^{9} + 22q^{10} - q^{11} + 24q^{12} + 16q^{13} + 18q^{14} + 33q^{15} + 32q^{16} + 9q^{17} + 36q^{18} - 38q^{19} + 44q^{20} + 27q^{21} - 2q^{22} - 4q^{23} + 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{273})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 38x + 88$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
19	$+1$

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	6	11	9	−1	16	9	−38	−4	−20	−238	−262	−122	−427	125	140	96	−551	−912	704	−321	−512	1188	1776	1772

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.127 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 engine's checks

dim M^+ at levels [1, 2, 3, 6, 19, 38, 57, 114]: [1, 2, 2, 5, 6, 17, 22, 64] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(114, 4)[5]   # 114.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 114:4 1000