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

Newform orbit 133.4.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 8q^{2} - 17q^{3} + 18q^{4} - 33q^{5} + 44q^{6} + 49q^{7} - 63q^{8} + 84q^{9} + 34q^{10} - 71q^{11} - 191q^{12} - 116q^{13} - 56q^{14} - 88q^{15} + 82q^{16} - 158q^{17} - 403q^{18} - 133q^{19} - 488q^{20} - 119q^{21} - 205q^{22} - 236q^{23} + 332q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 25x^{6} + 95x^{5} - 1285x^{4} - 7653x^{3} - 14615x^{2} - 10924x - 2372$

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
$\operatorname{tr} a_p$	−8	−17	−33	49	−71	−116	−158	−133	−236	−327	120	−309	−869	547	−335	−147	−1439	−191	−834	−389	−222	−291	−3488	−453	−1647

$\operatorname{tr} a_n$ 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)[3]   # 133.4.a.d

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

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