Atlas › Modular forms › Weight 4 › Level 114 › 114.4.a › 114.4.a.a

Newform orbit 114.4.a.a

$q$ -expansion

$q - 2q^{2} - 3q^{3} + 4q^{4} - 19q^{5} + 6q^{6} + 9q^{7} - 8q^{8} + 9q^{9} + 38q^{10} - 13q^{11} - 12q^{12} + 38q^{13} - 18q^{14} + 57q^{15} + 16q^{16} + 99q^{17} - 18q^{18} - 19q^{19} - 76q^{20} - 27q^{21} + 26q^{22} + 68q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x - 8$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
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 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
$\operatorname{tr} a_p$	−2	−3	−19	9	−13	38	99	−19	68	130	262	−296	−8	73	−271	−502	540	587	684	992	−507	980	−492	810	−1046

$\operatorname{tr} a_n$ 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 114, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.127 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, 3, 6, 19, 38, 57, 114]: [1, 2, 2, 5, 6, 17, 22, 64] (certified at ell = 2147483647)
T = [(5, 1), (7, 3)]; h of degree 8 = 1 x 8 (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(114, 4)[0]   # 114.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 114:4 1000