Atlas › Modular forms › Weight 2 › Level 281 › 281.2.a › 281.2.a.b

Newform orbit 281.2.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q - q^{2} + 4q^{3} + 23q^{4} + 2q^{5} + 5q^{6} + 16q^{7} - 6q^{8} + 16q^{9} - 4q^{10} + q^{11} + 2q^{12} + 9q^{13} - 5q^{14} + 9q^{15} + 29q^{16} - 6q^{17} - 14q^{18} + 28q^{19} - 5q^{20} - 7q^{21} - 10q^{22} + 7q^{23} + 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 3x^{15} - 60x^{14} + 187x^{13} + 1308x^{12} - 4255x^{11} - 12774x^{10} + 44656x^{9} + 56761x^{8} - 236033x^{7} - 86178x^{6} + 623316x^{5} - 91746x^{4} - 741544x^{3} + 315940x^{2} + 313385x - 182959$

Atkin–Lehner signs

$p$	$w_p$
281	$-1$

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
$\operatorname{tr} a_p$	−1	4	2	16	1	9	−6	28	7	−8	59	2	−14	1	23	−17	7	7	−3	12	−9	24	7	5	−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 0.17 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, 281]: [0, 24] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (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(281, 2)[1]   # 281.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 281:2 1000