Atlas › Modular forms › Weight 2 › Level 135 › 135.2.a › 135.2.a.d

Newform orbit 135.2.a.d

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{2} + 3q^{4} - 2q^{5} + 2q^{7} + 6q^{8} - q^{10} - 2q^{11} + 6q^{13} - 12q^{14} - 3q^{16} + 4q^{17} - 3q^{20} - 14q^{22} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 7x - 69$

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	−2	2	−2	6	4	0	6	−10	−4	4	2	−6	4	4	10	6	−16	−22	18	−16	−6	6	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 5, 9, 15, 27, 45, 135]: [0, 1, 1, 2, 4, 4, 8, 20] (certified at ell = 2147483629)
T = [(2, 1), (7, 3)]; h of degree 6 = 1 x 6 (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(135, 2)[3]   # 135.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 135:2 1000