Atlas › Modular forms › Weight 2 › Level 926 › 926.2.a › 926.2.a.d

Newform orbit 926.2.a.d

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{13} - 12x^{12} - 4x^{11} + 561x^{10} - 1805x^{9} - 5495x^{8} + 35870x^{7} - 23709x^{6} - 178903x^{5} + 392763x^{4} - 96698x^{3} - 481083x^{2} + 507848x - 146047$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
463	$+1$

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
$\operatorname{tr} a_p$	13	5	7	13	4	10	−10	16	11	−12	19	−8	−9	4	21	9	23	20	20	−20	−30	−11	34	−22	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 2, 463, 926]: [0, 1, 39, 118] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 5 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(926, 2)[3]   # 926.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 926:2 1000