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

Newform orbit 15.8.a.b

$q$ -expansion

$q - 13q^{2} - 27q^{3} + 41q^{4} - 125q^{5} + 351q^{6} + 1380q^{7} + 1131q^{8} + 729q^{9} + 1625q^{10} - 3304q^{11} - 1107q^{12} + 8506q^{13} - 17940q^{14} + 3375q^{15} - 19951q^{16} - 9994q^{17} - 9477q^{18} + 41236q^{19} - 5125q^{20} - 37260q^{21} + 42952q^{22} + 84120q^{23} - 30537q^{24} + O(q^{25})$

Coefficient field

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

$x - 1367$

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 even 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
$\operatorname{tr} a_p$	−13	−27	−125	1380	−3304	8506	−9994	41236	84120	132802	−55800	228170	−139670	−755492	836984	1641650	−989656	−1658162	−4523844	−389408	5617330	3901080	−9394116	2803746	5099426

$\operatorname{tr} a_n$ 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)[1]   # 15.8.a.b

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

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