Atlas › Modular forms › Weight 2 › Level 243 › 243.2.a › 243.2.a.f

Newform orbit 243.2.a.f

Trace form

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

$3q + 3q^{2} + 3q^{4} + 6q^{5} - 3q^{7} + 6q^{8} + 3q^{11} - 3q^{13} - 3q^{14} + 3q^{16} + 9q^{17} - 3q^{19} - 3q^{20} + 6q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - 27x^{2} - 2010x + 1979$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 243$ )

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	−3	3	−3	9	−3	6	12	−12	−3	−3	−12	−6	18	−21	6	6	−9	6	6	−6	0	15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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]: [0, 1, 2, 4, 10, 28] (certified at ell = 2147483179)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(243, 2)[5]   # 243.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 243:2 1000