Atlas › Modular forms › Weight 8 › Level 10 › 10.8.a › 10.8.a.a

Newform orbit 10.8.a.a

$q$ -expansion

$q + 8q^{2} + 28q^{3} + 64q^{4} + 125q^{5} + 224q^{6} + 104q^{7} + 512q^{8} - 1403q^{9} + 1000q^{10} - 5148q^{11} + 1792q^{12} - 8602q^{13} + 832q^{14} + 3500q^{15} + 4096q^{16} + 20274q^{17} - 11224q^{18} + 45500q^{19} + 8000q^{20} + 2912q^{21} - 41184q^{22} - 72072q^{23} + 14336q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{7}$ is a root of

$x - 132$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$

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
$\operatorname{tr} a_p$	8	28	125	104	−5148	−8602	20274	45500	−72072	231510	−80128	104654	584922	−795532	425664	1500798	246420	893942	−2336836	−203688	−3805702	5053040	−45492	980010	−5247646

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ for the primes $p < 1000$ not dividing 10, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 2, 5, 10]: [1, 3, 5, 13] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 1 = 1 x 1 (phi(ord eps) x dim S_k^new); 1 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(10, 8)[0]   # 10.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 10:8 1000