Atlas › Modular forms › Weight 2 › Level 297 › 297.2.a › 297.2.a.e

Newform orbit 297.2.a.e

Trace form

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

$2q - 2q^{2} + 4q^{4} - 2q^{5} - 4q^{7} - 12q^{8} - 4q^{10} - 2q^{11} - 4q^{13} + 10q^{14} + 16q^{16} - 2q^{17} + 8q^{20} + 2q^{22} - 16q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{2} + 108x + 2673$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 297$ )
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
$\operatorname{tr} a_p$	−2	0	−2	−4	−2	−4	−2	0	−16	6	8	−6	4	0	−10	−6	−10	−12	−2	0	−4	−8	0	18	10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 11, 27, 33, 99, 297]: [0, 1, 2, 2, 4, 6, 14, 38] (certified at ell = 2147482801)
T = [(2, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(297, 2)[4]   # 297.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 297:2 1000