Atlas › Modular forms › Weight 2 › Level 231 › 231.2.a › 231.2.a.e

Newform orbit 231.2.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} - 88x^{2} + 344x + 32609$

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
$\operatorname{tr} a_p$	2	3	4	−3	−3	−4	8	−8	10	−4	−2	0	14	−14	0	0	0	−6	−4	−12	−20	12	6	26	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 7, 11, 21, 33, 77, 231]: [0, 1, 1, 2, 4, 6, 10, 36] (certified at ell = 2147482951)
T = [(2, 1), (5, 3), (17, 5), (19, 7), (23, 9)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 3 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, 2)[4]   # 231.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 231:2 1000