Atlas › Modular forms › Weight 6 › Level 15 › 15.6.a › 15.6.a.a

Newform orbit 15.6.a.a

$q$ -expansion

$q - 2q^{2} - 9q^{3} - 28q^{4} - 25q^{5} + 18q^{6} - 132q^{7} + 120q^{8} + 81q^{9} + 50q^{10} + 472q^{11} + 252q^{12} - 686q^{13} + 264q^{14} + 225q^{15} + 656q^{16} - 1562q^{17} - 162q^{18} - 2180q^{19} + 700q^{20} + 1188q^{21} - 944q^{22} + 264q^{23} - 1080q^{24} + O(q^{25})$

Coefficient field

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

$x + 134$

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
$\operatorname{tr} a_p$	−2	−9	−25	−132	472	−686	−1562	−2180	264	170	7272	−142	−16198	−10316	18568	21514	34600	−35738	−5772	−69088	−70526	47640	74004	−90030	−33502

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.214 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, 3, 3, 12] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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, 6)[0]   # 15.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 15:6 1000