Atlas › Modular forms › Weight 6 › Level 43 › 43.6.a › 43.6.a.a

Newform orbit 43.6.a.a

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 12q^{2} - 26q^{3} + 122q^{4} - 212q^{5} - 69q^{6} - 136q^{7} - 666q^{8} + 546q^{9} - 617q^{10} - 532q^{11} - 4195q^{12} - 2492q^{13} - 4240q^{14} - 1780q^{15} + 1882q^{16} - 2534q^{17} - 3711q^{18} - 1678q^{19} - 2607q^{20} - 2256q^{21} + 11502q^{22} - 2488q^{23} + 19953q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 38x^{7} - 643x^{6} - 23394x^{5} + 75089x^{4} + 3165034x^{3} - 4442068x^{2} - 72841260x + 174248300$

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 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
$\operatorname{tr} a_p$	−12	−26	−212	−136	−532	−2492	−2534	−1678	−2488	−4360	5704	−3772	−10698	−14792	−77864	−62352	−26224	−82540	27784	−9504	14260	160248	−77176	−265692	144742

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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, 20] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 4 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, 6)[0]   # 43.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 43:6 1000