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

Newform orbit 133.4.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 3q^{2} + 7q^{3} + 29q^{4} - 29q^{5} + 34q^{6} - 42q^{7} + 90q^{8} + 47q^{9} + 70q^{10} + 197q^{11} + 65q^{12} - 22q^{13} - 21q^{14} - 16q^{15} + 289q^{16} + 36q^{17} + 24q^{18} - 114q^{19} - 128q^{20} - 49q^{21} + 319q^{22} - 80q^{23} + 556q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 10x^{5} - 127x^{4} + 1138x^{3} + 4369x^{2} - 29974x - 3828$

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
$\operatorname{tr} a_p$	3	7	−29	−42	197	−22	36	−114	−80	233	144	719	−927	71	79	1053	−255	−819	1054	157	204	663	3668	1049	1803

$\operatorname{tr} a_n$ 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)[2]   # 133.4.a.c

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

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