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

Newform orbit 95.6.a.d

Trace form

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

$9q + 4q^{2} + 9q^{3} + 164q^{4} - 225q^{5} - 98q^{6} + 313q^{7} - 48q^{8} + 1342q^{9} - 100q^{10} + 944q^{11} - 314q^{12} + 2389q^{13} + 502q^{14} - 225q^{15} + 6436q^{16} + 1953q^{17} + 5290q^{18} + 3249q^{19} - 4100q^{20} - 6143q^{21} - 8540q^{22} + 2829q^{23} + 17856q^{24} + O(q^{25})$

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

$x^{9} - 13x^{8} - 1808x^{7} + 14560x^{6} + 1086375x^{5} - 3082409x^{4} - 238100942x^{3} - 561317760x^{2} + 11448718324x + 45528268372$

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
$\operatorname{tr} a_p$	4	9	−225	313	944	2389	1953	3249	2829	−1061	8030	34338	14786	−10718	−6696	−21759	14223	113324	31825	148192	117263	143138	−79774	−20666	156262

$\operatorname{tr} a_n$ 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)[3]   # 95.6.a.d

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

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