Atlas › Modular forms › Weight 2 › Level 892 › 892.2.a › 892.2.a.e

Newform orbit 892.2.a.e

Trace form

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

$8q - q^{3} + 5q^{5} + 5q^{7} + q^{9} + 13q^{11} + 3q^{13} + 6q^{15} + q^{17} + 8q^{19} - 5q^{21} + 21q^{23} + O(q^{25})$

Coefficient field

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

$x^{8} - 14x^{7} - 162x^{6} + 2710x^{5} + 4242x^{4} - 142337x^{3} + 130758x^{2} + 2245652x - 4388504$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 892$ )
223	$+1$

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
$\operatorname{tr} a_p$	0	−1	5	5	13	3	1	8	21	11	13	6	−12	23	23	14	2	−13	9	38	−6	−12	28	−9	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 4, 223, 446, 892]: [0, 1, 2, 19, 58, 115] (certified at ell = 2147482591)
T = [(3, 1), (5, 3)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 3 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(892, 2)[4]   # 892.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 892:2 1000