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

Newform orbit 95.4.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{2} - 11q^{3} + q^{4} + 15q^{5} + 25q^{6} - 5q^{7} - 27q^{8} - 12q^{9} - 15q^{10} - 4q^{11} - 75q^{12} - 125q^{13} - 93q^{14} - 55q^{15} - 7q^{16} - 119q^{17} - 138q^{18} - 57q^{19} + 5q^{20} + 35q^{21} - 88q^{22} - 129q^{23} + 201q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 36x^{2} + 250x + 136$

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
$\operatorname{tr} a_p$	−3	−11	15	−5	−4	−125	−119	−57	−129	37	90	−250	−94	−210	524	−1417	15	−912	237	−124	−2457	−98	−454	−62	−1730

$\operatorname{tr} a_n$ 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)[4]   # 95.4.a.e

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

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