Atlas › Modular forms › Weight 2 › Level 363 › 363.2.a › 363.2.a.d

Newform orbit 363.2.a.d

Trace form

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

$2q - 3q^{2} - 2q^{3} + 3q^{4} + q^{5} + 3q^{6} - 2q^{7} - 6q^{8} + 2q^{9} + q^{10} - 3q^{12} - 4q^{13} + 3q^{14} - q^{15} + 13q^{16} - 9q^{17} - 3q^{18} + 5q^{19} - 6q^{20} + 2q^{21} - 4q^{23} + 6q^{24} + O(q^{25})$

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

$x^{2} - 20$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
11	— ( $11^2 \mid 363$ )

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	−2	1	−2	0	−4	−9	5	−4	−12	−1	−8	−4	0	−9	−3	17	−3	−3	5	2	22	−6	−12	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.318 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, 11, 33, 121, 363]: [0, 1, 2, 6, 12, 46] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(363, 2)[3]   # 363.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 363:2 1000