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

Newform orbit 135.4.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - q^{2} + 23q^{4} + 15q^{5} + 44q^{7} - 36q^{8} - 5q^{10} + 38q^{11} + 28q^{13} - 108q^{14} + 191q^{16} - 19q^{17} + 187q^{19} + 115q^{20} + 122q^{22} - 81q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{3} - 131x^{2} + 5137x - 58008$

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
$\operatorname{tr} a_p$	−1	0	15	44	38	28	−19	187	−81	160	227	78	−338	22	−472	521	140	595	878	−602	1294	629	−1287	2154	1392

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

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)[5]   # 135.4.a.f

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

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