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

Newform orbit 231.4.a.e

$q$ -expansion

$q + 5q^{2} + 3q^{3} + 17q^{4} - 6q^{5} + 15q^{6} + 7q^{7} + 45q^{8} + 9q^{9} - 30q^{10} - 11q^{11} + 51q^{12} + 70q^{13} + 35q^{14} - 18q^{15} + 89q^{16} + 126q^{17} + 45q^{18} - 80q^{19} - 102q^{20} + 21q^{21} - 55q^{22} - 200q^{23} + 135q^{24} + O(q^{25})$

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

$x + 1743$

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 even 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
$\operatorname{tr} a_p$	5	3	−6	7	−11	70	126	−80	−200	134	−244	−314	278	−372	−84	182	−756	694	820	160	−2	40	760	−102	−862

$\operatorname{tr} a_n$ 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)[4]   # 231.4.a.e

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

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