Atlas › Modular forms › Weight 4 › Level 129 › 129.4.a › 129.4.a.f

Newform orbit 129.4.a.f

Trace form

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

$8q + q^{2} + 24q^{3} + 55q^{4} + 4q^{5} + 3q^{6} + 52q^{7} + 33q^{8} + 72q^{9} + 56q^{10} + 66q^{11} + 165q^{12} + 94q^{13} - 252q^{14} + 12q^{15} + 259q^{16} - 38q^{17} + 9q^{18} + 36q^{19} - 174q^{20} + 156q^{21} + 450q^{22} + 204q^{23} + 99q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} - 5x^{7} - 883x^{6} + 6247x^{5} + 205478x^{4} - 1851974x^{3} - 6016856x^{2} + 43224384x + 108957696$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
43	$-1$

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
$\operatorname{tr} a_p$	1	24	4	52	66	94	−38	36	204	120	534	488	−150	344	−904	−546	−1258	1584	86	876	944	2396	86	−78	2242

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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, 3, 43, 129]: [1, 2, 12, 46] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (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(129, 4)[5]   # 129.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 129:4 1000