Atlas › Modular forms › Weight 4 › Level 142 › 142.4.a › 142.4.a.b

Newform orbit 142.4.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 8q^{2} - 14q^{3} + 16q^{4} - 21q^{5} - 28q^{6} - 47q^{7} + 32q^{8} + 2q^{9} - 42q^{10} - 92q^{11} - 56q^{12} - 123q^{13} - 94q^{14} - 92q^{15} + 64q^{16} - 258q^{17} + 4q^{18} - 62q^{19} - 84q^{20} + 149q^{21} - 184q^{22} + 103q^{23} - 112q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 35x^{3} + 293x^{2} - 328x - 5363$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
71	$+1$

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
$\operatorname{tr} a_p$	8	−14	−21	−47	−92	−123	−258	−62	103	353	−365	−137	108	220	−35	−600	−482	−132	686	−284	630	−1405	−1077	−702	1010

$\operatorname{tr} a_n$ 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, 2, 71, 142]: [1, 2, 19, 56] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (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(142, 4)[1]   # 142.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 142:4 1000