Atlas › Modular forms › Weight 2 › Level 636 › 636.2.a › 636.2.a.b

Newform orbit 636.2.a.b

Trace form

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

$2q - 2q^{3} + q^{5} - 3q^{7} + 2q^{9} - 6q^{11} - 3q^{13} - q^{15} + 2q^{17} - 10q^{19} + 3q^{21} - q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} + 8x + 3$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 636$ )
3	$+1$
53	$-1$

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	−2	1	−3	−6	−3	2	−10	−1	−2	−4	−13	−3	−9	4	2	−8	0	6	−9	−6	−10	27	18	−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 0.292 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, 3, 4, 6, 12, 53, 106, 159, 212, 318, 636]: [0, 1, 1, 2, 3, 5, 5, 15, 20, 30, 58, 114] (certified at ell = 2147483053)
T = [(5, 1), (7, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(636, 2)[1]   # 636.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 636:2 1000