Atlas › Modular forms › Weight 2 › Level 448 › 448.2.a › 448.2.a.j

Newform orbit 448.2.a.j

Trace form

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

$2q + 2q^{3} - 2q^{5} + 2q^{7} + 6q^{9} + 4q^{11} - 6q^{13} + 8q^{15} - 2q^{19} + 2q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = 8661T_{3} + 7879T_{5} + 8882T_{11} + 1822T_{13} + 7435T_{17} + 9474T_{19} + 6076T_{23} + 2656T_{29}$ , whose minimal polynomial is

$x^{2} - 55820x - 4568135920$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 448$ )
7	$-1$

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	−2	2	4	−6	0	−2	8	0	4	0	−8	−4	12	20	14	−18	−8	8	12	8	14	−12	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.704 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, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, 448]: [0, 1, 2, 1, 3, 4, 4, 7, 6, 12, 10, 20, 36, 68] (certified at ell = 2147483137)
T = [(3, 8661), (5, 7879), (11, 8882), (13, 1822), (17, 7435), (19, 9474), (23, 6076), (29, 2656)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 8 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(448, 2)[9]   # 448.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 448:2 1000