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

Newform orbit 71.4.a.c

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 2q^{2} + 7q^{3} + 54q^{4} + 15q^{5} + 42q^{6} + 48q^{7} + 24q^{8} + 161q^{9} + 110q^{10} + 70q^{11} + 96q^{12} + 166q^{13} + 104q^{14} + 72q^{15} + 354q^{16} + 228q^{17} - 91q^{18} + 159q^{19} - 541q^{20} - 32q^{21} - 412q^{22} - 284q^{23} - 499q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 9x^{11} - 319x^{10} + 2865x^{9} + 35482x^{8} - 330282x^{7} - 1505375x^{6} + 16327235x^{5} + 10071436x^{4} - 294235416x^{3} + 479866824x^{2} + 216446256x - 523980576$

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 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
$\operatorname{tr} a_p$	2	7	15	48	70	166	228	159	−284	−227	580	315	1044	213	−326	508	438	2328	1338	−852	3085	1963	−1229	1787	996

$\operatorname{tr} a_n$ 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)[2]   # 71.4.a.c

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

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