Atlas › Modular forms › Weight 2 › Level 181 › 181.2.a › 181.2.a.b

Newform orbit 181.2.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{9} - 6x^{8} - 15x^{7} + 136x^{6} - 75x^{5} - 641x^{4} + 775x^{3} + 578x^{2} - 862x + 73$

Atkin–Lehner signs

$p$	$w_p$
181	$-1$

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	3	1	2	24	−8	−1	4	14	13	−7	−20	10	−5	−2	−1	24	−16	9	21	−5	10	23	−1	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.161 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, 181]: [0, 15] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 2 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(181, 2)[1]   # 181.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 181:2 1000