Atlas › Modular forms › Weight 4 › Level 243 › 243.4.a › 243.4.a.k

Newform orbit 243.4.a.k

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 6q^{2} + 36q^{4} + 30q^{5} + 18q^{7} + 129q^{8} + 114q^{11} - 36q^{13} + 201q^{14} + 144q^{16} + 234q^{17} - 90q^{19} + 345q^{20} + 300q^{23} + O(q^{25})$

Coefficient field

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

$x^{9} - 984x^{8} + 105135x^{7} + 153499218x^{6} - 39247977468x^{5} - 5318971769244x^{4} + 1830079973170830x^{3} + 79445699767223853x^{2} - 22905472743992356518x - 1246504433750598374119$

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
$\operatorname{tr} a_p$	6	0	30	18	114	−36	234	−90	300	753	180	−252	768	−252	1176	1764	1191	−360	−252	90	−144	−441	312	−2106	−1386

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.235 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 engine's checks

dim M^+ at levels [1, 3, 9, 27, 81, 243]: [1, 2, 4, 10, 28, 82] (certified at ell = 2147483179)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 12 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, 4)[10]   # 243.4.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 243:4 1000