Atlas › Modular forms › Weight 4 › Level 158 › 158.4.a › 158.4.a.a

Newform orbit 158.4.a.a

$q$ -expansion

$q - 2q^{2} + 3q^{3} + 4q^{4} - 9q^{5} - 6q^{6} + 9q^{7} - 8q^{8} - 18q^{9} + 18q^{10} - 8q^{11} + 12q^{12} - 47q^{13} - 18q^{14} - 27q^{15} + 16q^{16} + 84q^{17} + 36q^{18} - 150q^{19} - 36q^{20} + 27q^{21} + 16q^{22} - 2q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 6$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
79	$-1$

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
$\operatorname{tr} a_p$	−2	3	−9	9	−8	−47	84	−150	−2	180	−328	−346	−268	328	−351	498	365	492	784	897	378	79	−1102	−1255	1129

$\operatorname{tr} a_n$ 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 158, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.122 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, 2, 79, 158]: [1, 2, 21, 62] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (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(158, 4)[0]   # 158.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 158:4 1000