Atlas › Modular forms › Weight 2 › Level 224 › 224.2.a › 224.2.a.d

Newform orbit 224.2.a.d

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 $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{2} - 70x - 1420$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 224$ )
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 0.226 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, 112, 224]: [0, 1, 2, 1, 3, 4, 4, 7, 6, 12, 20, 36] (certified at ell = 2147483497)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 6 = 1 x 6 (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(224, 2)[3]   # 224.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 224:2 1000