Atlas › Modular forms › Weight 2 › Level 578 › 578.2.a › 578.2.a.g

Newform orbit 578.2.a.g

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 3q^{2} - 3q^{3} + 3q^{4} - 6q^{5} - 3q^{6} - 3q^{7} + 3q^{8} - 6q^{10} - 6q^{11} - 3q^{12} - 12q^{13} - 3q^{14} + 3q^{15} + 3q^{16} - 6q^{19} - 6q^{20} - 6q^{21} - 6q^{22} + 3q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 21x^{2} + 126x + 179$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
17	— ( $17^2 \mid 578$ )

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
$\operatorname{tr} a_p$	3	−3	−6	−3	−6	−12	0	−6	3	−3	0	−6	−9	−9	6	3	3	−3	−6	27	−12	6	−3	−6	−39

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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, 17, 34, 289, 578]: [0, 1, 2, 6, 26, 78] (certified at ell = 2147483249)
T = [(3, 1), (5, 3)]; h of degree 23 = 1 x 23 (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(578, 2)[6]   # 578.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 578:2 1000