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

Newform orbit 41.8.a.a

Trace form

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

$10q - q^{2} - 68q^{3} + 383q^{4} - 320q^{5} - 810q^{6} - 2402q^{7} - 927q^{8} + 3450q^{9} - 7458q^{10} - 4122q^{11} - 2812q^{12} - 8184q^{13} - 35020q^{14} - 62394q^{15} - 61469q^{16} - 50096q^{17} - 163125q^{18} - 83680q^{19} - 188334q^{20} - 72432q^{21} - 68796q^{22} - 89516q^{23} - 303294q^{24} + O(q^{25})$

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

$x^{10} + 69x^{9} - 10301x^{8} - 430245x^{7} + 39105926x^{6} + 674518644x^{5} - 52479598612x^{4} - 344946178524x^{3} + 19187946246876x^{2} + 147581505587540x - 240107011681600$

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
$\operatorname{tr} a_p$	−1	−68	−320	−2402	−4122	−8184	−50096	−83680	−89516	−312452	−394640	−516236	689210	854120	703536	−397880	−2399348	−6529348	−6472598	−10883268	−8539644	−11382752	−6597908	−19052296	−3296972

$\operatorname{tr} a_n$ 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)[0]   # 41.8.a.a

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

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