Atlas › Modular forms › Weight 4 › Level 153 › 153.4.a › 153.4.a.a

Newform orbit 153.4.a.a

$q$ -expansion

$q - q^{2} - 7q^{4} + 10q^{5} - 8q^{7} + 15q^{8} - 10q^{10} - 12q^{11} - 26q^{13} + 8q^{14} + 41q^{16} - 17q^{17} - 148q^{19} - 70q^{20} + 12q^{22} - 152q^{23} + O(q^{25})$

Coefficient field

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

$x - 9$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 153$ )
17	$+1$

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
$\operatorname{tr} a_p$	−1	0	10	−8	−12	−26	−17	−148	−152	66	−32	−266	6	−92	288	546	−420	350	940	−424	378	288	−748	1558	530

$\operatorname{tr} a_n$ 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 153, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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, 17, 51, 153]: [1, 2, 4, 6, 20, 56] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 4 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(153, 4)[0]   # 153.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 153:4 1000