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

Newform orbit 926.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{15} - 2x^{14} - 90x^{13} + 145x^{12} + 3049x^{11} - 3983x^{10} - 49364x^{9} + 55473x^{8} + 407809x^{7} - 422215x^{6} - 1705912x^{5} + 1662629x^{4} + 3324548x^{3} - 2964245x^{2} - 2382900x + 1896500$

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
$\operatorname{tr} a_p$	−15	1	1	3	2	6	10	18	−1	2	19	16	25	14	−7	−5	−5	52	18	−8	26	43	−38	42	65

$\operatorname{tr} a_n$ 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)[4]   # 926.2.a.e

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

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