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

Newform orbit 181.2.a.a

Trace form

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

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

Coefficient field

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

$x^{5} + 8x^{4} + 21x^{3} + 20x^{2} + 4x - 1$

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 odd 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	−5	−5	−2	−20	−2	−5	−6	−6	−17	5	2	2	−5	4	3	−24	−20	13	−29	17	−6	−21	5	11

$\operatorname{tr} a_n$ 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)[0]   # 181.2.a.a

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

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