Atlas › Modular forms › Weight 8 › Level 43 › 43.8.a › 43.8.a.b

Newform orbit 43.8.a.b

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 110x^{12} - 14647x^{11} + 1644338x^{10} + 83647697x^{9} - 9040765014x^{8} - 238696796640x^{7} + 22147781872128x^{6} + 346543763033608x^{5} - 23407095660014208x^{4} - 213947978461547560x^{3} + 9098503326580908720x^{2} + 36787519140520780576x - 569654513316109811904$

Atkin–Lehner signs

$p$	$w_p$
43	$+1$

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
$\operatorname{tr} a_p$	16	94	998	1360	1620	13550	110880	105058	160184	285546	−99616	176038	−410260	−1033591	−424556	3992458	2248836	6210394	−1993648	4978064	8224814	6945708	22937328	9291302	10001852

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.273 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, 43]: [1, 26] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (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(43, 8)[1]   # 43.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 43:8 1000