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

Newform orbit 119.4.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 4q^{2} + 5q^{3} + 52q^{4} + 35q^{5} + 51q^{6} - 49q^{7} - 6q^{8} + 128q^{9} + 18q^{10} + 48q^{11} - 16q^{12} + 84q^{13} - 28q^{14} - 54q^{15} + 256q^{16} - 119q^{17} + 196q^{18} + 156q^{19} + 317q^{20} - 35q^{21} + 244q^{22} + 290q^{23} + 818q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - 9x^{6} - 223x^{5} + 1816x^{4} + 13472x^{3} - 105827x^{2} - 188571x + 1542267$

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
$\operatorname{tr} a_p$	4	5	35	−49	48	84	−119	156	290	1074	−291	596	649	257	−1108	321	−1176	1345	1379	2000	367	704	−16	1070	−39

$\operatorname{tr} a_n$ 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)[3]   # 119.4.a.d

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

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