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

Newform orbit 133.4.a.b

Trace form

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

$6q - 3q^{2} - 3q^{3} + 25q^{4} - 13q^{5} - 38q^{6} - 42q^{7} - 96q^{8} - q^{9} - 18q^{10} - 47q^{11} + 9q^{12} - 38q^{13} + 21q^{14} - 128q^{15} + 41q^{16} - 120q^{17} - 98q^{18} + 114q^{19} - 104q^{20} + 21q^{21} - 305q^{22} - 428q^{23} - 196q^{24} + O(q^{25})$

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

$x^{6} + 6x^{5} - 61x^{4} - 68x^{3} + 779x^{2} - 378x - 1196$

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
$\operatorname{tr} a_p$	−3	−3	−13	−42	−47	−38	−120	114	−428	−289	−96	617	55	−429	−589	−789	−265	−979	254	−757	−72	−287	−640	−641	−59

$\operatorname{tr} a_n$ 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)[1]   # 133.4.a.b

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

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