Atlas › Modular forms › Weight 2 › Level 933 › 933.2.a › 933.2.a.e

Newform orbit 933.2.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{14} - 10x^{13} - 6x^{12} + 347x^{11} - 795x^{10} - 3114x^{9} + 13026x^{8} + 929x^{7} - 57376x^{6} + 49124x^{5} + 78711x^{4} - 104521x^{3} - 22903x^{2} + 57561x - 11930$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
311	$+1$

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
$\operatorname{tr} a_p$	4	14	6	5	14	2	23	7	18	19	−4	−13	22	21	16	−2	22	−10	3	22	−14	−35	51	24	−5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.275 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, 3, 311, 933]: [0, 1, 27, 106] (certified at ell = 2147479121)
T = [(2, 1), (5, 1)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 6 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(933, 2)[4]   # 933.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 933:2 1000