Atlas › Modular forms › Weight 4 › Level 115 › 115.4.a › 115.4.a.a

Newform orbit 115.4.a.a

$q$ -expansion

$q + q^{2} + 4q^{3} - 7q^{4} - 5q^{5} + 4q^{6} - 32q^{7} - 15q^{8} - 11q^{9} - 5q^{10} + 40q^{11} - 28q^{12} - 66q^{13} - 32q^{14} - 20q^{15} + 41q^{16} + 130q^{17} - 11q^{18} - 88q^{19} + 35q^{20} - 128q^{21} + 40q^{22} + 23q^{23} - 60q^{24} + O(q^{25})$

Coefficient field

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

$x - 5$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
23	$-1$

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
$\operatorname{tr} a_p$	1	4	−5	−32	40	−66	130	−88	23	−130	40	−334	−22	−272	24	258	612	−366	−496	248	826	−296	−1296	−646	−1438

$\operatorname{tr} a_n$ 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 115, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 5, 23, 115]: [1, 3, 7, 38] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(115, 4)[0]   # 115.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 115:4 1000