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

Newform orbit 15.12.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - q^{2} + 729q^{3} + 4757q^{4} + 9375q^{5} - 243q^{6} - 14608q^{7} - 33999q^{8} + 177147q^{9} - 3125q^{10} + 540620q^{11} + 1155951q^{12} + 840970q^{13} + 5432712q^{14} + 2278125q^{15} + 5062961q^{16} + 15165038q^{17} - 59049q^{18} + 17743756q^{19} + 14865625q^{20} - 3549744q^{21} + 11176076q^{22} - 28140816q^{23} - 8261757q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{3} + 14609x^{2} - 2706663298x + 22551554333152$

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
$\operatorname{tr} a_p$	−1	729	9375	−14608	540620	840970	15165038	17743756	−28140816	−67382798	−206919496	−318337278	2110085854	418259692	−1599668584	4489142234	11102167484	−3568120958	2229942788	49842766696	40752219934	113159960	6259660308	−59972401554	−207831285882

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

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)[3]   # 15.12.a.d

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

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