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

Newform orbit 95.6.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 5q^{2} - 20q^{3} + 53q^{4} - 150q^{5} + 55q^{6} - 80q^{7} + 255q^{8} - 36q^{9} - 125q^{10} - 8q^{11} - 1365q^{12} - 390q^{13} - 2669q^{14} + 500q^{15} - 2479q^{16} - 310q^{17} - 2790q^{18} - 2166q^{19} - 1325q^{20} - 1300q^{21} - 4160q^{22} - 380q^{23} - 8007q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 15x^{5} - 776x^{4} - 5420x^{3} + 134596x^{2} + 241300x - 1043304$

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
$\operatorname{tr} a_p$	5	−20	−150	−80	−8	−390	−310	−2166	−380	−9364	−16280	−7490	−17348	−11380	4940	71710	−14830	−16244	−47080	−26388	−42370	51136	94600	103124	−360570

$\operatorname{tr} a_n$ 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)[2]   # 95.6.a.c

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

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