Atlas › Modular forms › Weight 4 › Level 182 › 182.4.a › 182.4.a.c

Newform orbit 182.4.a.c

$q$ -expansion

$q + 2q^{2} - 2q^{3} + 4q^{4} - 5q^{5} - 4q^{6} - 7q^{7} + 8q^{8} - 23q^{9} - 10q^{10} - 36q^{11} - 8q^{12} - 13q^{13} - 14q^{14} + 10q^{15} + 16q^{16} + 26q^{17} - 46q^{18} - 47q^{19} - 20q^{20} + 14q^{21} - 72q^{22} - 99q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17}$ is a root of

$x + 15$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	$+1$
13	$+1$

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	−2	−5	−7	−36	−13	26	−47	−99	−61	−23	−50	70	−19	191	195	264	310	−190	−166	873	−1191	259	−635	133

$\operatorname{tr} a_n$ 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 182, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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, 7, 13, 14, 26, 91, 182]: [1, 2, 3, 5, 8, 13, 30, 88] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (17, 7)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 7 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(182, 4)[2]   # 182.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 182:4 1000