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

Newform orbit 41.6.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 3q^{2} + 28q^{3} + 207q^{4} + 32q^{5} + 54q^{6} + 342q^{7} + 249q^{8} + 1194q^{9} + 102q^{10} + 846q^{11} + 4204q^{12} + 1504q^{13} + 3468q^{14} + 1966q^{15} + 5859q^{16} + 560q^{17} - 3713q^{18} + 4240q^{19} - 6182q^{20} - 3096q^{21} - 2628q^{22} - 1508q^{23} - 10166q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 31x^{9} - 1649x^{8} + 53623x^{7} + 868226x^{6} - 31370036x^{5} - 129178028x^{4} + 6695335084x^{3} - 9444577516x^{2} - 285838618620x + 548524184296$

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 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
$\operatorname{tr} a_p$	3	28	32	342	846	1504	560	4240	−1508	−124	10384	5524	16810	24160	58984	23456	52428	113540	85506	75236	−85148	178200	−125412	−62696	154548

$\operatorname{tr} a_n$ 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)[1]   # 41.6.a.b

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

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