Atlas › Modular forms › Weight 4 › Level 231 › 231.4.a › 231.4.a.b

Newform orbit 231.4.a.b

$q$ -expansion

$q - 2q^{2} + 3q^{3} - 4q^{4} + q^{5} - 6q^{6} - 7q^{7} + 24q^{8} + 9q^{9} - 2q^{10} - 11q^{11} - 12q^{12} + 7q^{13} + 14q^{14} + 3q^{15} - 16q^{16} - 14q^{17} - 18q^{18} - 45q^{19} - 4q^{20} - 21q^{21} + 22q^{22} - 88q^{23} + 72q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{17} + 7T_{19} + 9T_{23}$ is a root of

$x + 1176$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
7	$+1$
11	$+1$

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	3	1	−7	−11	7	−14	−45	−88	−69	22	57	−380	48	−385	−672	−469	−342	−139	132	145	1244	522	822	272

$\operatorname{tr} a_n$ 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 231, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.381 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, 7, 11, 21, 33, 77, 231]: [1, 2, 3, 4, 10, 14, 26, 100] (certified at ell = 2147482951)
T = [(2, 1), (5, 3), (17, 5), (19, 7), (23, 9)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 11 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(231, 4)[1]   # 231.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 231:4 1000