Embedded newform 16.12.e.a.13.2

• Label: 16.12.e.a.13.2
• Level: $16$
• Weight: $12$
• Character: 16.13
• Analytic conductor: $12.293$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2934908890\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-45.0084 - 4.71633i) q^{2} +(396.720 + 396.720i) q^{3} +(2003.51 + 424.549i) q^{4} +(7616.23 - 7616.23i) q^{5} +(-15984.7 - 19726.8i) q^{6} +16490.4i q^{7} +(-88172.6 - 28557.5i) q^{8} +137626. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−45.0084	−	4.71633i	−0.994555	−	0.104217i
							\(3\)	396.720	+	396.720i	0.942577	+	0.942577i	0.998439	−	0.0558611i	\(-0.0177904\pi\)
−0.0558611	+	0.998439i	\(0.517790\pi\)
\(5\)	7616.23	−	7616.23i	1.08995	−	1.08995i	0.0944131	−	0.995533i	\(-0.469903\pi\)
							0.995533	−	0.0944131i	\(-0.0300974\pi\)
\(7\)			16490.4i			0.370844i	0.982659	+	0.185422i	\(0.0593651\pi\)
							−0.982659	+	0.185422i	\(0.940635\pi\)
\(11\)	568615.	−	568615.i	1.06453	−	1.06453i	0.0667634	−	0.997769i	\(-0.478733\pi\)
							0.997769	−	0.0667634i	\(-0.0212673\pi\)
\(13\)	−1.11150e6	−	1.11150e6i	−0.830272	−	0.830272i	0.157282	−	0.987554i	\(-0.449727\pi\)
							−0.987554	+	0.157282i	\(0.949727\pi\)
\(17\)	6.34711e6			1.08419			0.542097	−	0.840316i	\(-0.317631\pi\)
							0.542097	+	0.840316i	\(0.317631\pi\)
\(19\)	3.59606e6	+	3.59606e6i	0.333182	+	0.333182i	0.853794	−	0.520611i	\(-0.174296\pi\)
							−0.520611	+	0.853794i	\(0.674296\pi\)
\(23\)			4.79362e7i			1.55296i	0.630141	+	0.776480i	\(0.282997\pi\)
							−0.630141	+	0.776480i	\(0.717003\pi\)
\(29\)	−2.69107e7	−	2.69107e7i	−0.243633	−	0.243633i	0.574719	−	0.818351i	\(-0.305112\pi\)
							−0.818351	+	0.574719i	\(0.805112\pi\)
\(31\)	1.46392e8			0.918395			0.459197	−	0.888334i	\(-0.348137\pi\)
							0.459197	+	0.888334i	\(0.348137\pi\)
\(37\)	−4.69927e8	+	4.69927e8i	−1.11409	+	1.11409i	−0.121501	+	0.992591i	\(0.538771\pi\)
							−0.992591	+	0.121501i	\(0.961229\pi\)
\(41\)			3.61263e8i			0.486981i	0.969903	+	0.243491i	\(0.0782925\pi\)
							−0.969903	+	0.243491i	\(0.921707\pi\)
\(43\)	5.36452e8	−	5.36452e8i	0.556486	−	0.556486i	−0.371819	−	0.928305i	\(-0.621266\pi\)
							0.928305	+	0.371819i	\(0.121266\pi\)
\(47\)	−1.46204e9			−0.929866			−0.464933	−	0.885346i	\(-0.653922\pi\)
							−0.464933	+	0.885346i	\(0.653922\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.13.2	yes	42
4.3	odd	2		64.12.e.a.17.4		42
8.3	odd	2		128.12.e.a.33.18		42
8.5	even	2		128.12.e.b.33.4		42
16.3	odd	4		128.12.e.a.97.18		42
16.5	even	4	inner	16.12.e.a.5.2	✓	42
16.11	odd	4		64.12.e.a.49.4		42
16.13	even	4		128.12.e.b.97.4		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.2	✓	42	16.5	even	4	inner
16.12.e.a.13.2	yes	42	1.1	even	1	trivial
64.12.e.a.17.4		42	4.3	odd	2
64.12.e.a.49.4		42	16.11	odd	4
128.12.e.a.33.18		42	8.3	odd	2
128.12.e.a.97.18		42	16.3	odd	4
128.12.e.b.33.4		42	8.5	even	2
128.12.e.b.97.4		42	16.13	even	4