Atlas › Modular forms › Weight 2 › Level 333 › 333.2.a › 333.2.a.f

Newform orbit 333.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 4q^{4} + 8q^{7} + 8q^{13} + 4q^{16} + 8q^{19} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13} + 11T_{17}$ , whose minimal polynomial is

$x^{4} - 112x^{3} + 3930x^{2} - 44464x + 83683$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 333$ )
37	$-1$

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	0	0	8	0	8	0	8	0	0	8	4	0	8	0	0	0	8	8	0	−16	8	0	0	8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 37, 111, 333]: [0, 1, 2, 3, 14, 40] (certified at ell = 2147483629)
T = [(2, 1), (5, 3), (7, 5), (11, 7), (13, 9), (17, 11)]; h of degree 15 = 1 x 15 (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(333, 2)[5]   # 333.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 333:2 1000