Atlas › Modular forms › Weight 4 › Level 118 › 118.4.a › 118.4.a.a

Newform orbit 118.4.a.a

$q$ -expansion

$q - 2q^{2} + 5q^{3} + 4q^{4} - 5q^{5} - 10q^{6} - 33q^{7} - 8q^{8} - 2q^{9} + 10q^{10} - 4q^{11} + 20q^{12} - 30q^{13} + 66q^{14} - 25q^{15} + 16q^{16} - 14q^{17} + 4q^{18} + 97q^{19} - 20q^{20} - 165q^{21} + 8q^{22} - 134q^{23} - 40q^{24} + O(q^{25})$

Coefficient field

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

$x$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
59	$-1$

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	5	−5	−33	−4	−30	−14	97	−134	1	−28	290	−5	192	−326	−537	59	472	856	168	−686	−919	362	−312	−514

$\operatorname{tr} a_n$ 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 118, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 59, 118]: [1, 2, 16, 47] (certified at ell = 2147482481)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 3 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(118, 4)[0]   # 118.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 118:4 1000