Atlas › Modular forms › Weight 6 › Level 95 › 95.6.a › 95.6.a.b

Newform orbit 95.6.a.b

Trace form

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

$5q - 7q^{2} - 16q^{3} + 81q^{4} + 125q^{5} - 273q^{6} - 312q^{7} - 135q^{8} + 359q^{9} - 175q^{10} + 140q^{11} - 667q^{12} - 1436q^{13} + 845q^{14} - 400q^{15} + 1217q^{16} - 3592q^{17} + 4524q^{18} + 1805q^{19} + 2025q^{20} - 3764q^{21} - 4664q^{22} - 5836q^{23} - 8295q^{24} + O(q^{25})$

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

$x^{5} + 23x^{4} - 370x^{3} - 6720x^{2} + 44204x + 352232$

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 odd 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
$\operatorname{tr} a_p$	−7	−16	125	−312	140	−1436	−3592	1805	−5836	−4090	−9984	−5212	1010	−30936	−30972	−28676	76698	−15266	35168	110972	−153516	91864	2004	−127990	−218392

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.266 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, 10, 52] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 6 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, 6)[1]   # 95.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 95:6 1000