Atlas › Modular forms › Weight 2 › Level 263 › 263.2.a › 263.2.a.b

Newform orbit 263.2.a.b

Trace form

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

$17q + q^{2} + 7q^{3} + 19q^{4} + 3q^{5} - 3q^{6} + 7q^{7} + 3q^{8} + 26q^{9} + 9q^{10} - 2q^{11} + 13q^{12} + 27q^{13} - 11q^{14} - 3q^{15} + 15q^{16} + 16q^{17} - 12q^{18} + 5q^{19} - 15q^{20} - 5q^{21} - 4q^{22} - 4q^{23} - 28q^{24} + O(q^{25})$

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

$x^{17} - 8x^{16} - 30x^{15} + 369x^{14} + 38x^{13} - 6440x^{12} + 7592x^{11} + 51788x^{10} - 101882x^{9} - 175819x^{8} + 514051x^{7} + 69056x^{6} - 914265x^{5} + 580216x^{4} - 18608x^{3} - 30312x^{2} - 2837x - 67$

Atkin–Lehner signs

$p$	$w_p$
263	$-1$

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	7	3	7	−2	27	16	5	−4	−12	11	17	2	9	3	7	−17	1	19	−17	61	1	−16	10	114

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.171 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, 263]: [0, 23] (certified at ell = 2147482739)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (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(263, 2)[1]   # 263.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 263:2 1000