Atlas › Modular forms › Weight 2 › Level 639 › 639.2.a › 639.2.a.k

Newform orbit 639.2.a.k

Trace form

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

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

Coefficient field

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

$x^{4} - 24x^{3} + 196x^{2} - 624x + 624$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 639$ )
71	$-1$

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	0	8	2	10	4	14	0	10	−4	−8	−14	24	2	4	12	−6	−6	−4	4	16	−2	4	16	−18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 71, 213, 639]: [0, 1, 2, 7, 26, 74] (certified at ell = 2147482681)
T = [(2, 1), (5, 3)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 4 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(639, 2)[10]   # 639.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 639:2 1000