Atlas › Modular forms › Weight 2 › Level 729 › 729.2.a › 729.2.a.a

Newform orbit 729.2.a.a

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 3q^{2} + 3q^{4} - 6q^{5} - 6q^{8} + 3q^{10} - 12q^{11} - 6q^{14} - 3q^{16} - 9q^{17} + 3q^{19} - 6q^{20} + 6q^{22} - 15q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{6} + 21x^{5} + 96x^{4} - 525x^{3} - 4941x^{2} - 11682x - 8691$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 729$ )

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	0	−6	0	−12	0	−9	3	−15	−12	0	3	−15	0	−21	−9	−24	−9	−9	−27	−6	0	−12	−9	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.234 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, 27, 81, 243, 729]: [0, 1, 2, 4, 10, 28, 82] (certified at ell = 2147481883)
T = [(2, 1), (5, 3)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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(729, 2)[0]   # 729.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 729:2 1000