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

Newform orbit 142.4.a.d

Trace form

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

$5q + 10q^{2} + 10q^{3} + 20q^{4} + 29q^{5} + 20q^{6} + 23q^{7} + 40q^{8} + 47q^{9} + 58q^{10} + 40q^{11} + 40q^{12} + 33q^{13} + 46q^{14} + 88q^{15} + 80q^{16} + 48q^{17} + 94q^{18} + 110q^{19} + 116q^{20} - 99q^{21} + 80q^{22} + 33q^{23} + 80q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 39x^{4} + 259x^{3} + 5254x^{2} - 55215x - 14370$

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 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
$\operatorname{tr} a_p$	10	10	29	23	40	33	48	110	33	35	139	−163	−606	−272	−11	−510	−18	−224	−344	355	−516	−417	975	−76	122

$\operatorname{tr} a_n$ 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)[3]   # 142.4.a.d

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

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