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

Newform orbit 133.4.a.a

$q$ -expansion

$q + 4q^{2} + 8q^{3} + 8q^{4} + 6q^{5} + 32q^{6} - 7q^{7} + 37q^{9} + 24q^{10} - 68q^{11} + 64q^{12} + 8q^{13} - 28q^{14} + 48q^{15} - 64q^{16} + 14q^{17} + 148q^{18} - 19q^{19} + 48q^{20} - 56q^{21} - 272q^{22} + 188q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x - 12$

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 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
$\operatorname{tr} a_p$	4	8	6	−7	−68	8	14	−19	188	70	252	−186	192	488	−216	178	−500	−298	494	−618	−842	10	228	600	−976

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 133, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 133.4.a.a

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

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