Atlas › Modular forms › Weight 4 › Level 119 › 119.4.a › 119.4.a.e

Newform orbit 119.4.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 2q^{2} + 11q^{3} + 38q^{4} - 3q^{5} + 9q^{6} + 63q^{7} + 24q^{8} + 74q^{9} + 134q^{10} - 8q^{11} + 56q^{12} + 164q^{13} + 14q^{14} + 34q^{15} + 178q^{16} + 153q^{17} + 98q^{18} + 244q^{19} - 41q^{20} + 77q^{21} - 80q^{22} - 14q^{23} + 298q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 13x^{8} - 138x^{7} + 1941x^{6} + 6333x^{5} - 88583x^{4} - 148807x^{3} + 1123722x^{2} + 2670975x + 1377081$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
17	$-1$

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
$\operatorname{tr} a_p$	2	11	−3	63	−8	164	153	244	−14	−234	555	−364	−45	−135	−172	101	280	639	35	−1616	1049	2304	2508	2762	3107

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 7, 17, 119]: [1, 3, 6, 38] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 4 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(119, 4)[4]   # 119.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 119:4 1000