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

Newform orbit 95.4.a.a

$q$ -expansion

$q + 4q^{3} - 8q^{4} - 5q^{5} - 22q^{7} - 11q^{9} - 12q^{11} - 32q^{12} + 8q^{13} - 20q^{15} + 64q^{16} - 66q^{17} + 19q^{19} + 40q^{20} - 88q^{21} - 30q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3}$ is a root of

$x - 12$

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
$\operatorname{tr} a_p$	0	4	−5	−22	−12	8	−66	19	−30	−6	−64	−16	54	182	594	396	−564	−706	−628	−984	14	−328	−294	918	−1564

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 95, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 95.4.a.a

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

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