Atlas › Modular forms › Weight 4 › Level 133 › 133.4.a › 133.4.a.e

Newform orbit 133.4.a.e

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 2q^{2} + 13q^{3} + 30q^{4} + 37q^{5} - 12q^{6} + 56q^{7} - 9q^{8} + 89q^{9} + 42q^{10} + 105q^{11} + 73q^{12} + 16q^{13} + 14q^{14} - 8q^{15} + 122q^{16} + 284q^{17} - 133q^{18} + 152q^{19} + 208q^{20} + 91q^{21} - 117q^{22} + 332q^{23} + 156q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 15x^{7} - 75x^{6} + 1735x^{5} - 593x^{4} - 59221x^{3} + 109120x^{2} + 628964x - 1584848$

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 even 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
$\operatorname{tr} a_p$	2	13	37	56	105	16	284	152	332	9	124	−105	333	527	93	−1223	579	503	−1292	915	1052	−683	1400	117	−969

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.112 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, 7, 19, 133]: [1, 3, 6, 42] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (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(133, 4)[4]   # 133.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 133:4 1000