Atlas › Modular forms › Weight 12 › Level 15 › 15.12.a › 15.12.a.a

Newform orbit 15.12.a.a

$q$ -expansion

$q - 56q^{2} - 243q^{3} + 1088q^{4} + 3125q^{5} + 13608q^{6} + 27984q^{7} + 53760q^{8} + 59049q^{9} - 175000q^{10} - 112028q^{11} - 264384q^{12} - 1096922q^{13} - 1567104q^{14} - 759375q^{15} - 5238784q^{16} - 249566q^{17} - 3306744q^{18} - 13712420q^{19} + 3400000q^{20} - 6800112q^{21} + 6273568q^{22} + 41395728q^{23} - 13063680q^{24} + O(q^{25})$

Coefficient field

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

$x - 27928$

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
$\operatorname{tr} a_p$	−56	−243	3125	27984	−112028	−1096922	−249566	−13712420	41395728	−4533850	−265339008	−212136946	−1266969958	14129548	−2657273336	2402699278	7498737220	−4064828858	6871514244	−13283734648	−28875844262	27100302240	−34365255132	−63500412630	19634495234

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

Sato–Tate

The normalized $a_p / 2p^{11/2}$ 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 0.335 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]: [2, 5, 7, 24] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 5 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, 12)[0]   # 15.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 15:12 1000