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

Newform orbit 142.4.a.a

Trace form

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

$4q - 8q^{2} - 8q^{3} + 16q^{4} - 11q^{5} + 16q^{6} + 19q^{7} - 32q^{8} + 34q^{9} + 22q^{10} - 4q^{11} - 32q^{12} - 29q^{13} - 38q^{14} - 70q^{15} + 64q^{16} - 164q^{17} - 68q^{18} - 48q^{19} - 44q^{20} - 201q^{21} + 8q^{22} - 383q^{23} + 64q^{24} + O(q^{25})$

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

$x^{4} + 19x^{3} + 53x^{2} - 660x - 2935$

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	−8	−11	19	−4	−29	−164	−48	−383	−553	−117	81	−318	226	−183	−362	384	824	840	284	−546	53	−1441	−2758	−1258

$\operatorname{tr} a_n$ 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)[0]   # 142.4.a.a

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

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