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

Newform orbit 933.2.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} + 13x^{9} + 44x^{8} - 61x^{7} - 575x^{6} - 684x^{5} + 1160x^{4} + 2968x^{3} + 1538x^{2} - 402x - 293$

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 odd 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
$\operatorname{tr} a_p$	−7	10	−6	−14	−12	−13	−26	−17	−11	−32	−2	3	−33	−23	4	−16	−19	16	−1	17	1	−15	−67	−10	9

$\operatorname{tr} a_n$ 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)[2]   # 933.2.a.c

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

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