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

Newform orbit 243.4.a.a

$q$ -expansion

$q - 3q^{2} + q^{4} + 3q^{5} - 10q^{7} + 21q^{8} - 9q^{10} - 12q^{11} + 8q^{13} + 30q^{14} - 71q^{16} + 126q^{17} - 91q^{19} + 3q^{20} + 36q^{22} + 93q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ is a root of

$x + 128$

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	3	−10	−12	8	126	−91	93	−183	170	−172	−474	−172	159	−603	−564	−430	−439	351	−727	1232	−498	1044	1151

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 243, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 243.4.a.a

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

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