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

Newform orbit 95.4.a.g

Trace form

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

$6q - q^{2} + 5q^{3} + 25q^{4} - 30q^{5} + 61q^{6} + 5q^{7} - 3q^{8} + 135q^{9} + 5q^{10} + 8q^{11} + 35q^{12} + 79q^{13} + 151q^{14} - 25q^{15} + 285q^{16} + 35q^{17} - 48q^{18} - 114q^{19} - 125q^{20} + 403q^{21} + 320q^{22} - 111q^{23} + 365q^{24} + O(q^{25})$

Coefficient field

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

$x^{6} - 14x^{5} - 1458x^{4} + 13206x^{3} + 603493x^{2} - 2106538x - 59431232$

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
$\operatorname{tr} a_p$	−1	5	−30	5	8	79	35	−114	−111	107	574	490	−256	−212	674	−1729	−25	1626	741	792	1529	62	−80	4256	1910

$\operatorname{tr} a_n$ 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)[6]   # 95.4.a.g

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

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