Atlas › Modular forms › Weight 10 › Level 5 › 5.10.a › 5.10.a.a

Newform orbit 5.10.a.a

$q$ -expansion

$q - 8q^{2} - 114q^{3} - 448q^{4} - 625q^{5} + 912q^{6} + 4242q^{7} + 7680q^{8} - 6687q^{9} + 5000q^{10} - 46208q^{11} + 51072q^{12} - 115934q^{13} - 33936q^{14} + 71250q^{15} + 167936q^{16} + 494842q^{17} + 53496q^{18} - 1008740q^{19} + 280000q^{20} - 483588q^{21} + 369664q^{22} - 532554q^{23} - 875520q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 122$

Atkin–Lehner signs

$p$	$w_p$
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 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
$\operatorname{tr} a_p$	−8	−114	−625	4242	−46208	−115934	494842	−1008740	−532554	4196390	−3365028	−14931358	11056262	−6396794	−35559158	39738586	−85185620	45748642	−45286158	−189967468	412170946	95040840	261706326	−19938630	−19503358

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

Sato–Tate

The normalized $a_p / 2p^{9/2}$ for the primes $p < 1000$ not dividing 5, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 5]: [1, 5] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (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(5, 10)[0]   # 5.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 5:10 1000