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

Newform orbit 71.6.a.a

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - 4q^{2} - 20q^{3} + 122q^{4} - 72q^{5} - 306q^{6} - 344q^{7} - 192q^{8} + 101q^{9} - 850q^{10} - 356q^{11} - 960q^{12} - 2122q^{13} - 1264q^{14} - 648q^{15} + 674q^{16} - 2610q^{17} - 5390q^{18} - 6816q^{19} - 13078q^{20} - 8202q^{21} - 8806q^{22} - 6980q^{23} - 17478q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 24x^{10} - 1030x^{9} - 16930x^{8} + 383751x^{7} + 2709140x^{6} - 42552857x^{5} - 157493648x^{4} + 1534157675x^{3} + 1869051958x^{2} - 15245941404x + 13837652904$

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
$\operatorname{tr} a_p$	−4	−20	−72	−344	−356	−2122	−2610	−6816	−6980	−5752	−14206	8216	−27942	−1936	27802	17602	−10732	−71000	1138	−55451	−239248	−154544	−78708	−235304	−369972

$\operatorname{tr} a_n$ 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)[0]   # 71.6.a.a

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

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