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

Newform orbit 95.6.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 9q^{2} + 38q^{3} + 209q^{4} + 225q^{5} + 91q^{6} + 80q^{7} + 447q^{8} + 1047q^{9} + 225q^{10} + 624q^{11} + 1925q^{12} + 762q^{13} - 1353q^{14} + 950q^{15} + 1773q^{16} + 2188q^{17} - 1170q^{18} - 3249q^{19} + 5225q^{20} + 5968q^{21} + 8056q^{22} + 944q^{23} + 2909q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 47x^{8} - 852x^{7} + 56656x^{6} + 127157x^{5} - 22660641x^{4} + 31500410x^{3} + 3401007992x^{2} - 4654753356x - 148052592456$

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
$\operatorname{tr} a_p$	9	38	225	80	624	762	2188	−3249	944	6346	4244	17190	−150	−7896	1340	31914	46230	2794	−48082	−98152	96448	105656	158072	23446	142510

$\operatorname{tr} a_n$ 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)[4]   # 95.6.a.e

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

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