Atlas › Modular forms › Weight 12 › Level 16 › 16.12.a › 16.12.a.d

Newform orbit 16.12.a.d

Trace form

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

$2q - 56q^{3} + 7868q^{5} - 91056q^{7} + 540202q^{9} - 159080q^{11} + 1050476q^{13} + 10494832q^{15} + 1430884q^{17} + 21866600q^{19} + 40052544q^{21} - 35806736q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{109})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 7812x - 60195580$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 16$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	−56	7868	−91056	−159080	1050476	1430884	21866600	−35806736	−228827700	−64722112	75558780	1201214196	45519832	1229079264	−3808549924	6012926584	9789792908	−14703095224	−4319991088	11055639476	−51957623264	108227975912	71188291860	−1699807676

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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, 2, 4, 8, 16]: [2, 4, 7, 13, 24] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 3 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(16, 12)[3]   # 16.12.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 16:12 1000