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

Newform orbit 129.4.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 2q^{2} + 6q^{3} + 6q^{4} - 14q^{5} - 6q^{6} - 38q^{7} - 30q^{8} + 18q^{9} - 46q^{10} + 14q^{11} + 18q^{12} - 74q^{13} + 78q^{14} - 42q^{15} - 78q^{16} + 108q^{17} - 18q^{18} + 6q^{19} + 78q^{20} - 114q^{21} + 6q^{22} - 228q^{23} - 90q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{10})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 16x + 24$

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
$\operatorname{tr} a_p$	−2	6	−14	−38	14	−74	108	6	−228	−186	−312	−280	−336	−86	198	400	−252	−936	372	652	1268	−1608	1434	112	−1074

$\operatorname{tr} a_n$ 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)[2]   # 129.4.a.c

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

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