Atlas › Modular forms › Weight 8 › Level 41 › 41.8.a › 41.8.a.b

Newform orbit 41.8.a.b

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 15q^{2} + 40q^{3} + 1151q^{4} + 180q^{5} + 486q^{6} + 2400q^{7} + 2145q^{8} + 12198q^{9} + 542q^{10} + 2048q^{11} - 10130q^{12} + 10800q^{13} + 52472q^{14} + 56896q^{15} + 160083q^{16} + 31820q^{17} + 235075q^{18} + 88808q^{19} + 223710q^{20} + 139008q^{21} + 216590q^{22} + 182280q^{23} + 142618q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 55x^{13} - 21853x^{12} + 952695x^{11} + 182466162x^{10} - 6227922310x^{9} - 742930945668x^{8} + 19166332130180x^{7} + 1567989571683480x^{6} - 27127638718934720x^{5} - 1665923954462977716x^{4} + 12903019010882456340x^{3} + 745301155823493375208x^{2} + 1816753746218525801400x - 41940721579072236102576$

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
$\operatorname{tr} a_p$	15	40	180	2400	2048	10800	31820	88808	182280	−175984	677152	346980	−964894	295200	−1392840	−1850560	3372168	5479380	10917760	7769000	8689860	16493240	−6140920	4370788	−16968300

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.275 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, 26] (certified at ell = 2147482921)
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(41, 8)[1]   # 41.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 41:8 1000