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

Newform orbit 129.4.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 3q^{2} - 15q^{3} + 19q^{4} - 12q^{5} + 9q^{6} - 18q^{7} - 15q^{8} + 45q^{9} - 32q^{10} - 60q^{11} - 57q^{12} + 42q^{13} - 246q^{14} + 36q^{15} - 93q^{16} - 72q^{17} - 27q^{18} - 102q^{19} - 420q^{20} + 54q^{21} - 86q^{22} - 414q^{23} + 45q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 15x^{4} - 242x^{3} - 3618x^{2} + 8656x + 128496$

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 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	−15	−12	−18	−60	42	−72	−102	−414	−588	−370	242	60	215	−354	−264	−1500	498	186	−1680	−250	−1568	−840	−1782	1726

$\operatorname{tr} a_n$ 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)[3]   # 129.4.a.d

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

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