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

Newform orbit 141.4.a.e

Trace form

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

$7q - 3q^{2} - 21q^{3} + 35q^{4} + 6q^{5} + 9q^{6} + 30q^{7} - 57q^{8} + 63q^{9} + 64q^{11} - 105q^{12} + 84q^{13} + 10q^{14} - 18q^{15} + 499q^{16} - 58q^{17} - 27q^{18} + 362q^{19} + 406q^{20} - 90q^{21} + 308q^{22} - 18q^{23} + 171q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} - 3x^{6} - 672x^{5} + 3282x^{4} + 141579x^{3} - 864453x^{2} - 9055660x + 60233310$

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
$\operatorname{tr} a_p$	−3	−21	6	30	64	84	−58	362	−18	242	352	−332	−74	38	−329	−1346	1320	150	1382	−440	982	3866	1468	−786	1744

$\operatorname{tr} a_n$ 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)[4]   # 141.4.a.e

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

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