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

Newform orbit 231.4.a.l

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 5q^{2} + 15q^{3} + 21q^{4} + 7q^{5} + 15q^{6} - 35q^{7} + 60q^{8} + 45q^{9} + 55q^{10} + 55q^{11} + 63q^{12} + 111q^{13} - 35q^{14} + 21q^{15} + 201q^{16} + 136q^{17} + 45q^{18} + 111q^{19} + 219q^{20} - 105q^{21} + 55q^{22} - 28q^{23} + 180q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{17} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{5} - 1231x^{4} - 1678554x^{3} + 1647501531x^{2} + 576611076645x - 353026826363736$

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	15	7	−35	55	111	136	111	−28	61	−280	−41	426	424	75	1500	757	658	−583	−764	875	−244	924	−1110	−852

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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)[11]   # 231.4.a.l

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

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