Atlas › Modular forms › Weight 4 › Level 115 › 115.4.a › 115.4.a.e

Newform orbit 115.4.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 6q^{2} + 4q^{3} + 22q^{4} - 25q^{5} + 19q^{6} - 3q^{7} + 138q^{8} + 77q^{9} - 30q^{10} + 23q^{11} + 47q^{12} + 132q^{13} + 93q^{14} - 20q^{15} + 282q^{16} + 23q^{17} - 15q^{18} - 161q^{19} - 110q^{20} - 60q^{21} + 193q^{22} - 115q^{23} + 105q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} - 10x^{4} - 106x^{3} + 825x^{2} + 2930x - 15634$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
23	$+1$

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
$\operatorname{tr} a_p$	6	4	−25	−3	23	132	23	−161	−115	401	32	−38	−12	−566	919	1156	1324	−1673	558	−108	1173	656	−82	570	633

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 5, 23, 115]: [1, 3, 7, 38] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(115, 4)[4]   # 115.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 115:4 1000