Atlas › Modular forms › Weight 4 › Level 129 › 129.4.a › 129.4.a.a

Newform orbit 129.4.a.a

$q$ -expansion

$q - q^{2} + 3q^{3} - 7q^{4} - 2q^{5} - 3q^{6} + 6q^{7} + 15q^{8} + 9q^{9} + 2q^{10} - 48q^{11} - 21q^{12} - 62q^{13} - 6q^{14} - 6q^{15} + 41q^{16} - 66q^{17} - 9q^{18} - 92q^{19} + 14q^{20} + 18q^{21} + 48q^{22} + 106q^{23} + 45q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 3$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
43	$+1$

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	3	−2	6	−48	−62	−66	−92	106	−18	−196	0	502	−43	74	−40	744	752	36	−224	−1006	376	732	−1334	−242

$\operatorname{tr} a_n$ 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 129, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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, 3, 43, 129]: [1, 2, 12, 46] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (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(129, 4)[0]   # 129.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 129:4 1000