Atlas › Modular forms › Weight 10 › Level 15 › 15.10.a › 15.10.a.a

Newform orbit 15.10.a.a

$q$ -expansion

$q - 4q^{2} + 81q^{3} - 496q^{4} + 625q^{5} - 324q^{6} - 7680q^{7} + 4032q^{8} + 6561q^{9} - 2500q^{10} - 86404q^{11} - 40176q^{12} - 149978q^{13} + 30720q^{14} + 50625q^{15} + 237824q^{16} - 207622q^{17} - 26244q^{18} + 716284q^{19} - 310000q^{20} - 622080q^{21} + 345616q^{22} + 1369920q^{23} + 326592q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{7}$ is a root of

$x + 7684$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
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 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
$\operatorname{tr} a_p$	−4	81	625	−7680	−86404	−149978	−207622	716284	1369920	−3194402	−2349000	18735710	−29282630	−1516724	615752	4747430	60616076	−126745682	−111182652	−175551608	−61233350	234431160	118910388	−316534326	242912258

$\operatorname{tr} a_n$ 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 15, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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, 3, 5, 15]: [1, 4, 5, 20] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 6 = 1 x 6 (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(15, 10)[0]   # 15.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 15:10 1000