Atlas › Modular forms › Weight 4 › Level 58 › 58.4.a › 58.4.a.b

Newform orbit 58.4.a.b

$q$ -expansion

$q + 2q^{2} - 7q^{3} + 4q^{4} - 15q^{5} - 14q^{6} - 18q^{7} + 8q^{8} + 22q^{9} - 30q^{10} + 27q^{11} - 28q^{12} - 57q^{13} - 36q^{14} + 105q^{15} + 16q^{16} - 44q^{17} + 44q^{18} + 152q^{19} - 60q^{20} + 126q^{21} + 54q^{22} - 152q^{23} - 56q^{24} + O(q^{25})$

Coefficient field

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

$x + 22$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
29	$+1$

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	−7	−15	−18	27	−57	−44	152	−152	−29	−173	−120	−314	339	−357	−59	−572	−420	660	726	1004	361	−168	58	−1206

$\operatorname{tr} a_n$ 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 58, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 29, 58]: [1, 2, 9, 25] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(58, 4)[1]   # 58.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 58:4 1000