Atlas › Modular forms › Weight 4 › Level 95 › 95.4.a › 95.4.a.f

Newform orbit 95.4.a.f

Trace form

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

$5q - 3q^{2} - 4q^{3} + 15q^{4} + 25q^{5} - 44q^{6} + 72q^{7} - 21q^{8} + 45q^{9} - 15q^{10} + 88q^{11} + 66q^{12} + 62q^{13} + 48q^{14} - 20q^{15} + 195q^{16} + 194q^{17} + 73q^{18} + 95q^{19} + 75q^{20} - 64q^{21} + 216q^{22} + 224q^{23} - 84q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 15x^{4} - 593x^{3} - 5131x^{2} + 27496x + 139454$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
19	$-1$

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
$\operatorname{tr} a_p$	−3	−4	25	72	88	62	194	95	224	−474	−128	6	58	756	576	166	−1308	1002	652	−600	1786	480	−292	−1782	2242

$\operatorname{tr} a_n$ 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, 19, 95]: [1, 3, 6, 32] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 18 = 1 x 18 (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(95, 4)[5]   # 95.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 95:4 1000