Atlas › Modular forms › Weight 4 › Level 141 › 141.4.a › 141.4.a.a

Newform orbit 141.4.a.a

Trace form

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

$2q + 8q^{2} + 6q^{3} + 16q^{4} + 6q^{5} + 24q^{6} + 20q^{7} + 18q^{9} + 24q^{10} + 84q^{11} + 48q^{12} + 12q^{13} + 80q^{14} + 18q^{15} - 128q^{16} - 60q^{17} + 72q^{18} - 38q^{19} + 48q^{20} + 60q^{21} + 336q^{22} - 236q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{111})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 14x - 62$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
47	$-1$

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
$\operatorname{tr} a_p$	8	6	6	20	84	12	−60	−38	−236	98	−198	−84	166	102	94	−856	552	−68	−430	1404	−752	388	100	944	−1548

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 47, 141]: [1, 2, 13, 50] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (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(141, 4)[0]   # 141.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 141:4 1000