Atlas › Modular forms › Weight 8 › Level 15 › 15.8.a › 15.8.a.a

Newform orbit 15.8.a.a

$q$ -expansion

$q - 22q^{2} + 27q^{3} + 356q^{4} - 125q^{5} - 594q^{6} - 420q^{7} - 5016q^{8} + 729q^{9} + 2750q^{10} - 2944q^{11} + 9612q^{12} - 11006q^{13} + 9240q^{14} - 3375q^{15} + 64784q^{16} - 16546q^{17} - 16038q^{18} - 25364q^{19} - 44500q^{20} - 11340q^{21} + 64768q^{22} - 5880q^{23} - 135432q^{24} + O(q^{25})$

Coefficient field

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

$x + 442$

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	27	−125	−420	−2944	−11006	−16546	−25364	−5880	163042	−201600	120530	−115910	−19148	841016	501890	−1586176	−372962	4561044	1512832	−1522910	4231920	−1854204	−6888174	3700034

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.255 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, 5, 16] (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, 8)[0]   # 15.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 15:8 1000