Atlas › Modular forms › Weight 2 › Level 881 › 881.2.a › 881.2.a.a

Newform orbit 881.2.a.a

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q - 5q^{2} - 9q^{3} + 11q^{4} - 6q^{5} - 5q^{6} - 27q^{7} - 9q^{8} + 10q^{9} - 15q^{10} - 16q^{11} - 10q^{12} - 13q^{13} - 7q^{14} - 21q^{15} - 13q^{16} - 5q^{17} - 9q^{18} - 35q^{19} - 10q^{20} - 15q^{21} - 21q^{22} - 18q^{23} - 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} + 14x^{26} + 25x^{25} - 476x^{24} - 2087x^{23} + 5466x^{22} + 42262x^{21} - 11954x^{20} - 424830x^{19} - 267903x^{18} + 2465291x^{17} + 2693838x^{16} - 8779983x^{15} - 11651306x^{14} + 19826505x^{13} + 27244341x^{12} - 29067872x^{11} - 34830382x^{10} + 27868388x^{9} + 22061548x^{8} - 16072488x^{7} - 5180310x^{6} + 3903688x^{5} + 338283x^{4} - 302033x^{3} - 12616x^{2} + 7246x + 331$

Atkin–Lehner signs

$p$	$w_p$
881	$+1$

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
$\operatorname{tr} a_p$	−5	−9	−6	−27	−16	−13	−5	−35	−18	−22	−62	−17	−25	−33	−6	−2	−1	−45	−14	−49	−19	−142	16	3	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 881]: [0, 74] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 8 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(881, 2)[0]   # 881.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 881:2 1000