Atlas › Modular forms › Weight 4 › Level 119 › 119.4.a › 119.4.a.a

Newform orbit 119.4.a.a

$q$ -expansion

$q - q^{2} - 6q^{3} - 7q^{4} - 20q^{5} + 6q^{6} - 7q^{7} + 15q^{8} + 9q^{9} + 20q^{10} + 60q^{11} + 42q^{12} - 68q^{13} + 7q^{14} + 120q^{15} + 41q^{16} - 17q^{17} - 9q^{18} - 70q^{19} + 140q^{20} + 42q^{21} - 60q^{22} - 176q^{23} - 90q^{24} + O(q^{25})$

Coefficient field

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

$x + 7$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
17	$+1$

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 even 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
$\operatorname{tr} a_p$	−1	−6	−20	−7	60	−68	−17	−70	−176	−90	196	22	−138	328	−12	−234	−54	44	−596	200	1122	480	−838	778	1142

$\operatorname{tr} a_n$ 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 119, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 7, 17, 119]: [1, 3, 6, 38] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (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(119, 4)[0]   # 119.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 119:4 1000