Atlas › Modular forms › Weight 4 › Level 147 › 147.4.a › 147.4.a.f

Newform orbit 147.4.a.f

$q$ -expansion

$q + 4q^{2} - 3q^{3} + 8q^{4} - 18q^{5} - 12q^{6} + 9q^{9} - 72q^{10} - 50q^{11} - 24q^{12} + 36q^{13} + 54q^{15} - 64q^{16} - 126q^{17} + 36q^{18} + 72q^{19} - 144q^{20} - 200q^{22} + 14q^{23} + O(q^{25})$

Coefficient field

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

$x + 50$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	— ( $7^2 \mid 147$ )

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
$\operatorname{tr} a_p$	4	−3	−18	0	−50	36	−126	72	14	158	36	−162	270	−324	72	−22	−468	−792	232	−734	−180	236	−36	−234	−468

$\operatorname{tr} a_n$ 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 147, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.132 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, 7, 21, 49, 147]: [1, 2, 3, 10, 15, 58] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 20 = 1 x 20 (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(147, 4)[5]   # 147.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 147:4 1000