Atlas › Modular forms › Weight 4 › Level 71 › 71.4.a › 71.4.a.b

Newform orbit 71.4.a.b

Trace form

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

$4q - 3q^{2} - 6q^{3} + 5q^{4} - 9q^{5} - 31q^{6} - 35q^{7} - 9q^{8} - 2q^{9} - 34q^{10} - 64q^{11} - 41q^{12} - 101q^{13} - 7q^{14} - 32q^{15} - 135q^{16} - 150q^{17} + 126q^{18} - 12q^{19} + 18q^{20} - 35q^{21} + 138q^{22} + 321q^{23} + 297q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 9x^{3} - 111x - 118$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	−3	−6	−9	−35	−64	−101	−150	−12	321	−83	−321	−97	114	−158	−443	578	−558	−1506	428	284	−1494	611	1247	−354	−378

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 71]: [1, 19] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 3 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(71, 4)[1]   # 71.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 71:4 1000