Atlas › Modular forms › Weight 2 › Level 999 › 999.2.a › 999.2.a.f

Newform orbit 999.2.a.f

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{6} + 15x^{5} - 22x^{4} - 569x^{3} - 501x^{2} + 2202x + 1497$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 999$ )
37	$-1$

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
$\operatorname{tr} a_p$	−3	0	−4	−1	−7	−2	−10	1	−4	−22	−8	6	−18	−6	−8	−3	−21	−9	5	−3	0	−21	3	−29	−29

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.522 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, 27, 37, 111, 333, 999]: [0, 1, 2, 4, 3, 14, 40, 116] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 48 = 1 x 48 (phi(ord eps) x dim S_k^new); 7 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(999, 2)[5]   # 999.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 999:2 1000