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

Newform orbit 639.2.a.f

Trace form

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

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

Coefficient field

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

$x^{2} - 18x + 76$

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
$\operatorname{tr} a_p$	3	0	5	−4	8	−1	4	−8	3	−3	8	−1	17	−15	−15	−9	−4	10	−5	−2	2	−5	12	14	5

$\operatorname{tr} a_n$ 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)[5]   # 639.2.a.f

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

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