Atlas › Modular forms › Weight 6 › Level 71 › 71.6.a › 71.6.a.b

Newform orbit 71.6.a.b

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 4q^{2} + 16q^{3} + 346q^{4} + 128q^{5} + 126q^{6} + 244q^{7} + 192q^{8} + 1802q^{9} + 750q^{10} + 854q^{11} + 768q^{12} + 1258q^{13} + 304q^{14} + 1152q^{15} + 7842q^{16} + 2592q^{17} + 8567q^{18} + 8712q^{19} + 19723q^{20} + 12472q^{21} + 13072q^{22} + 5304q^{23} + 20933q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{18} - 20x^{17} - 3475x^{16} + 55689x^{15} + 5077140x^{14} - 59891592x^{13} - 4094666944x^{12} + 30287285719x^{11} + 1995247417901x^{10} - 6113372814109x^{9} - 594106892145206x^{8} - 572969120151493x^{7} + 101827025468568239x^{6} + 469839682394350254x^{5} - 8263202611846021008x^{4} - 68766056795622286584x^{3} + 114794575258393730880x^{2} + 2510100582181768929600x + 6366234786642469224000$

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
$\operatorname{tr} a_p$	4	16	128	244	854	1258	2592	8712	5304	1804	12940	29304	18528	−10336	−28520	−31226	−12794	105870	32514	90738	178648	199868	260564	239728	459920

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 31] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 6 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, 6)[1]   # 71.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 71:6 1000