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

Newform orbit 15.10.a.b

$q$ -expansion

$q + 22q^{2} - 81q^{3} - 28q^{4} - 625q^{5} - 1782q^{6} - 5988q^{7} - 11880q^{8} + 6561q^{9} - 13750q^{10} - 14648q^{11} + 2268q^{12} + 37906q^{13} - 131736q^{14} + 50625q^{15} - 247024q^{16} - 441098q^{17} + 144342q^{18} + 441820q^{19} + 17500q^{20} + 485028q^{21} - 322256q^{22} + 2264136q^{23} + 962280q^{24} + O(q^{25})$

Coefficient field

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

$x + 5966$

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
$\operatorname{tr} a_p$	22	−81	−625	−5988	−14648	37906	−441098	441820	2264136	−1049350	−7910568	−20992558	13285562	−23130764	−13873688	−57635174	−32042120	110664022	−118568268	276679712	−264023294	448202760	851015796	189894930	−1014149278

$\operatorname{tr} a_n$ 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)[1]   # 15.10.a.b

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

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