Atlas › Modular forms › Weight 6 › Level 41 › 41.6.a › 41.6.a.a

Newform orbit 41.6.a.a

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 5q^{2} - 8q^{3} + 15q^{4} - 68q^{5} - 162q^{6} - 344q^{7} - 135q^{8} + 222q^{9} - 698q^{10} - 1504q^{11} - 2122q^{12} - 1016q^{13} - 1200q^{14} - 3144q^{15} - 813q^{16} + 372q^{17} + 1407q^{18} - 1800q^{19} + 2662q^{20} - 2856q^{21} + 334q^{22} - 2544q^{23} + 7266q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 13x^{5} - 697x^{4} - 8365x^{3} + 37398x^{2} + 89666x - 289816$

Atkin–Lehner signs

$p$	$w_p$
41	$+1$

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
$\operatorname{tr} a_p$	−5	−8	−68	−344	−1504	−1016	372	−1800	−2544	6096	−18688	−15324	−10086	−29464	−18512	−16344	3272	20044	−23808	−12320	−70316	−136048	5704	200508	72436

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 41]: [1, 18] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (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(41, 6)[0]   # 41.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 41:6 1000