Atlas › Modular forms › Weight 2 › Level 873 › 873.2.a › 873.2.a.f

Newform orbit 873.2.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{2} + 29x + 59$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 873$ )
97	$-1$

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	4	−7	−1	−9	−1	−8	−6	7	−5	−6	16	−5	−18	−6	4	−16	11	4	10	2	12	−9	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.427 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, 97, 291, 873]: [0, 1, 2, 8, 34, 100] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 9 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(873, 2)[5]   # 873.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 873:2 1000