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

Newform orbit 231.4.a.i

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 3q^{2} + 6q^{3} + 7q^{4} + 2q^{5} + 9q^{6} + 14q^{7} + 42q^{8} + 18q^{9} + 77q^{10} - 22q^{11} + 21q^{12} - 134q^{13} + 21q^{14} + 6q^{15} + 7q^{16} - 26q^{17} + 27q^{18} + 152q^{19} + 229q^{20} + 42q^{21} - 33q^{22} + 178q^{23} + 126q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{37})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{17} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{2} - 2545x + 1455633$

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
$\operatorname{tr} a_p$	3	6	2	14	−22	−134	−26	152	178	190	346	−206	244	242	920	−550	−412	252	16	−572	−1026	−52	138	1512	−1106

$\operatorname{tr} a_n$ 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)[8]   # 231.4.a.i

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

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