Atlas › Modular forms › Weight 2 › Level 992 › 992.2.a › 992.2.a.f

Newform orbit 992.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 4q^{3} + 2q^{5} + 4q^{7} + 4q^{9} + 10q^{11} + 2q^{15} + 2q^{17} + 8q^{19} - 2q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ , whose minimal polynomial is

$x^{4} - 144x^{3} - 1684x^{2} + 866736x - 25386656$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 992$ )
31	$-1$

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	4	2	4	10	0	2	8	2	0	4	−10	2	24	0	−2	20	0	24	−16	8	−6	6	−18	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.819 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, 8, 16, 31, 32, 62, 124, 248, 496, 992]: [0, 1, 2, 3, 4, 3, 6, 10, 19, 36, 68, 132] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 30 = 1 x 30 (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(992, 2)[5]   # 992.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 992:2 1000