Atlas › Modular forms › Weight 2 › Level 927 › 927.2.a › 927.2.a.g

Newform orbit 927.2.a.g

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{8} - 4x^{7} - 262x^{6} + 344x^{5} + 16474x^{4} + 20179x^{3} - 229418x^{2} - 478500x + 35656$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 927$ )
103	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	0	1	6	−6	9	4	16	11	0	17	−6	−12	9	6	16	−11	5	5	10	14	14	23	14	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 0.284 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, 9, 103, 309, 927]: [0, 1, 2, 9, 36, 106] (certified at ell = 2147482501)
T = [(2, 1), (5, 3)]; h of degree 43 = 1 x 43 (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(927, 2)[6]   # 927.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 927:2 1000