Atlas › Modular forms › Weight 4 › Level 135 › 135.4.a › 135.4.a.a

Newform orbit 135.4.a.a

$q$ -expansion

$q - 2q^{2} - 4q^{4} + 5q^{5} + 24q^{8} - 10q^{10} + 10q^{11} - 80q^{13} - 16q^{16} + 7q^{17} - 113q^{19} - 20q^{20} - 20q^{22} - 81q^{23} + O(q^{25})$

Coefficient field

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

$x + 2$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 135$ )
5	$-1$

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
$\operatorname{tr} a_p$	−2	0	5	0	10	−80	7	−113	−81	−220	−189	170	−130	10	160	631	−560	229	750	890	−890	−27	429	−750	−1480

$\operatorname{tr} a_n$ 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 135, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.129 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, 5, 9, 15, 27, 45, 135]: [1, 2, 3, 4, 8, 10, 20, 56] (certified at ell = 2147483629)
T = [(2, 1), (7, 3)]; h of degree 16 = 1 x 16 (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(135, 4)[0]   # 135.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 135:4 1000