Atlas › Modular forms › Weight 2 › Level 369 › 369.2.a › 369.2.a.f

Newform orbit 369.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + q^{2} + 5q^{4} + 2q^{5} + 6q^{7} + 9q^{8} - 10q^{10} - 2q^{11} - 2q^{13} - 4q^{14} + 13q^{16} + 6q^{17} + 4q^{19} - 6q^{20} - 4q^{22} - 4q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - 7x^{2} - 5x + 43$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 369$ )
41	$+1$

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
$\operatorname{tr} a_p$	1	0	2	6	−2	−2	6	4	−4	6	16	−6	−3	−4	0	−6	8	2	−2	−20	−2	32	0	6	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 0.197 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, 41, 123, 369]: [0, 1, 2, 4, 16, 44] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 16 = 1 x 16 (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(369, 2)[5]   # 369.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 369:2 1000