Atlas › Modular forms › Weight 4 › Level 181 › 181.4.a › 181.4.a.a

Newform orbit 181.4.a.a

$q$ -expansion

$q - 3q^{2} + 7q^{3} + q^{4} - 18q^{5} - 21q^{6} + 20q^{7} + 21q^{8} + 22q^{9} + 54q^{10} - 33q^{11} + 7q^{12} + 20q^{13} - 60q^{14} - 126q^{15} - 71q^{16} - 102q^{17} - 66q^{18} - 34q^{19} - 18q^{20} + 140q^{21} + 99q^{22} + 3q^{23} + 147q^{24} + O(q^{25})$

Coefficient field

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

$x - 4$

Atkin–Lehner signs

$p$	$w_p$
181	$-1$

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
$\operatorname{tr} a_p$	−3	7	−18	20	−33	20	−102	−34	3	−216	−133	−376	264	263	−93	−462	−279	155	−52	477	−943	992	1122	744	−1222

$\operatorname{tr} a_n$ 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 181, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.132 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, 181]: [1, 47] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(181, 4)[0]   # 181.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 181:4 1000