Atlas › Modular forms › Weight 4 › Level 118 › 118.4.a › 118.4.a.e

Newform orbit 118.4.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 8q^{2} + 2q^{3} + 16q^{4} + 16q^{5} - 4q^{6} + q^{7} - 32q^{8} + 58q^{9} - 32q^{10} - 17q^{11} + 8q^{12} + 3q^{13} - 2q^{14} + 90q^{15} + 64q^{16} + 305q^{17} - 116q^{18} + 8q^{19} + 64q^{20} + 330q^{21} + 34q^{22} + 322q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} - 18x^{3} - 300x^{2} + 4840x + 1888$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
59	$+1$

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	16	1	−17	3	305	8	322	470	46	−59	−175	−179	−134	862	−236	−584	−314	−585	−1118	−1265	2073	1888	−318

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 59, 118]: [1, 2, 16, 47] (certified at ell = 2147482481)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 3 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(118, 4)[4]   # 118.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 118:4 1000