Atlas › Modular forms › Weight 2 › Level 133 › 133.2.a › 133.2.a.b

Newform orbit 133.2.a.b

Trace form

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

$2q - q^{2} - 3q^{3} + 3q^{4} - 6q^{5} - 5q^{6} + 2q^{7} - 6q^{8} + 5q^{9} + 3q^{10} - 5q^{11} + 2q^{12} - 4q^{13} - q^{14} + 9q^{15} - 3q^{16} - 7q^{17} + 17q^{18} + 2q^{19} - 9q^{20} - 3q^{21} - 4q^{22} - 6q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} + 10x + 12$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
19	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

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	−3	−6	2	−5	−4	−7	2	−6	9	−1	0	5	−20	−2	−3	2	6	7	10	−15	8	−15	−14	12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.17 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, 7, 19, 133]: [0, 1, 2, 14] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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(133, 2)[1]   # 133.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 133:2 1000