Atlas › Modular forms › Weight 2 › Level 440 › 440.2.a › 440.2.a.f

Newform orbit 440.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19} + 11T_{23}$ , whose minimal polynomial is

$x^{2} - 216x + 11052$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 440$ )
5	$+1$
11	$-1$

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	1	−2	3	2	2	7	9	6	−5	5	−1	−20	2	−6	−13	−2	−3	0	−9	18	10	20	−1	26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.434 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, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440]: [0, 1, 2, 1, 3, 3, 2, 6, 5, 10, 9, 8, 16, 22, 42, 80] (certified at ell = 2147482921)
T = [(3, 1), (7, 3), (13, 5), (17, 7), (19, 9), (23, 11)]; h of degree 10 = 1 x 10 (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(440, 2)[5]   # 440.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 440:2 1000