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

Newform orbit 15.6.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - q^{2} - 18q^{3} + 141q^{4} + 50q^{5} + 9q^{6} - 112q^{7} - 243q^{8} + 162q^{9} - 25q^{10} + 248q^{11} - 1269q^{12} + 876q^{13} - 3216q^{14} - 450q^{15} + 3585q^{16} + 2036q^{17} - 81q^{18} + 1464q^{19} + 3525q^{20} + 1008q^{21} - 6668q^{22} - 3216q^{23} + 2187q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{409})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} + 113x - 19814$

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	−18	50	−112	248	876	2036	1464	−3216	1948	2672	8668	−7628	−16440	−19360	−14356	−904	20220	−12904	−40976	59124	107600	−122088	103764	−24764

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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)[2]   # 15.6.a.c

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

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