Atlas › Modular forms › Weight 4 › Level 27 › 27.4.a › 27.4.a.b

Newform orbit 27.4.a.b

$q$ -expansion

$q + 3q^{2} + q^{4} + 15q^{5} - 25q^{7} - 21q^{8} + 45q^{10} - 15q^{11} + 20q^{13} - 75q^{14} - 71q^{16} + 72q^{17} + 2q^{19} + 15q^{20} - 45q^{22} + 114q^{23} + O(q^{25})$

Coefficient field

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

$x - 18$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 27$ )

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	15	−25	−15	20	72	2	114	30	101	−430	−30	110	−330	621	−660	−376	−250	−360	785	488	489	−450	−1105

$\operatorname{tr} a_n$ 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 27, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.097 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]: [1, 2, 4, 10] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(27, 4)[1]   # 27.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 27:4 1000