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

Newform orbit 115.4.a.f

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 6q^{2} + 42q^{4} + 40q^{5} + 11q^{7} + 3q^{8} + 158q^{9} + 30q^{10} + 41q^{11} + 48q^{12} + 28q^{13} + 161q^{14} + 98q^{16} + 71q^{17} + 84q^{18} + 177q^{19} + 210q^{20} + 292q^{21} + 171q^{22} + 184q^{23} + 177q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 6x^{7} - 222x^{6} + 1242x^{5} + 14956x^{4} - 75868x^{3} - 291419x^{2} + 1173478x + 1062270$

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	0	40	11	41	28	71	177	184	225	−36	−348	620	−390	123	1406	−676	1447	−1582	1396	17	708	1486	1360	−855

$\operatorname{tr} a_n$ 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)[5]   # 115.4.a.f

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

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