Atlas › Modular forms › Weight 2 › Level 251 › 251.2.a › 251.2.a.b

Newform orbit 251.2.a.b

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 2q^{2} + 26q^{4} + 3q^{5} + q^{6} + 3q^{7} + 6q^{8} + 25q^{9} + 7q^{10} - q^{11} - 9q^{12} + 22q^{13} - 7q^{14} - 8q^{15} + 40q^{16} - q^{17} - 7q^{18} + 13q^{19} - 14q^{20} + 25q^{21} + 4q^{22} - 2q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

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

$x^{17} - 2x^{16} - 67x^{15} + 147x^{14} + 1702x^{13} - 3905x^{12} - 21240x^{11} + 49907x^{10} + 139820x^{9} - 338765x^{8} - 470516x^{7} + 1243843x^{6} + 649934x^{5} - 2319751x^{4} + 79259x^{3} + 1729437x^{2} - 623343x - 115664$

Atkin–Lehner signs

$p$	$w_p$
251	$-1$

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	0	3	3	−1	22	−1	13	−2	28	12	27	−1	9	−20	1	−20	59	15	−26	8	33	0	11	−10

$\operatorname{tr} a_n$ 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, 251]: [0, 22] (certified at ell = 2147482501)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (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(251, 2)[1]   # 251.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 251:2 1000