Embedded newform 16.22.e.a.5.9

• Label: 16.22.e.a.5.9
• Level: $16$
• Weight: $22$
• Character: 16.5
• Analytic conductor: $44.716$
• Analytic rank: $0$
• Dimension: $82$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(44.7163750859\)
Analytic rank:	\(0\)
Dimension:	\(82\)
Relative dimension:	\(41\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.9
Character	\(\chi\)	\(=\)	16.5
Dual form			16.22.e.a.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1166.53 - 858.118i) q^{2} +(132197. - 132197. i) q^{3} +(624420. + 2.00204e6i) q^{4} +(2.49441e7 + 2.49441e7i) q^{5} +(-2.67652e8 + 4.07709e7i) q^{6} +1.02743e9i q^{7} +(9.89578e8 - 2.87126e9i) q^{8} -2.44918e10i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1166.53	−	858.118i	−0.805527	−	0.592559i
							\(3\)	132197.	−	132197.i	1.29255	−	1.29255i	0.359352	−	0.933202i	\(-0.382998\pi\)
0.933202	−	0.359352i	\(-0.117002\pi\)
\(5\)	2.49441e7	+	2.49441e7i	1.14231	+	1.14231i	0.988027	+	0.154281i	\(0.0493062\pi\)
							0.154281	+	0.988027i	\(0.450694\pi\)
\(7\)			1.02743e9i			1.37475i	0.726301	+	0.687377i	\(0.241238\pi\)
							−0.726301	+	0.687377i	\(0.758762\pi\)
\(11\)	2.32385e10	+	2.32385e10i	0.270137	+	0.270137i	0.829155	−	0.559018i	\(-0.188822\pi\)
							−0.559018	+	0.829155i	\(0.688822\pi\)
\(13\)	−4.22461e11	+	4.22461e11i	−0.849927	+	0.849927i	−0.990124	−	0.140197i	\(-0.955227\pi\)
							0.140197	+	0.990124i	\(0.455227\pi\)
\(17\)	−3.16552e12			−0.380830			−0.190415	−	0.981704i	\(-0.560983\pi\)
							−0.190415	+	0.981704i	\(0.560983\pi\)
\(19\)	1.27298e13	−	1.27298e13i	0.476330	−	0.476330i	−0.427626	−	0.903956i	\(-0.640650\pi\)
							0.903956	+	0.427626i	\(0.140650\pi\)
\(23\)			7.31825e13i			0.368354i	0.982893	+	0.184177i	\(0.0589619\pi\)
							−0.982893	+	0.184177i	\(0.941038\pi\)
\(29\)	−2.75207e15	+	2.75207e15i	−1.21473	+	1.21473i	−0.245282	+	0.969452i	\(0.578881\pi\)
							−0.969452	+	0.245282i	\(0.921119\pi\)
\(31\)	5.30737e15			1.16301			0.581503	−	0.813544i	\(-0.302465\pi\)
							0.581503	+	0.813544i	\(0.302465\pi\)
\(37\)	5.57034e14	+	5.57034e14i	0.0190442	+	0.0190442i	0.716565	−	0.697521i	\(-0.245713\pi\)
							−0.697521	+	0.716565i	\(0.745713\pi\)
\(41\)			1.32562e17i			1.54237i	0.636609	+	0.771187i	\(0.280337\pi\)
							−0.636609	+	0.771187i	\(0.719663\pi\)
\(43\)	5.76677e16	+	5.76677e16i	0.406924	+	0.406924i	0.880665	−	0.473740i	\(-0.157096\pi\)
							−0.473740	+	0.880665i	\(0.657096\pi\)
\(47\)	4.22023e17			1.17033			0.585165	−	0.810914i	\(-0.301030\pi\)
							0.585165	+	0.810914i	\(0.301030\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.e.a.5.9	✓	82
4.3	odd	2		64.22.e.a.49.2		82
16.3	odd	4		64.22.e.a.17.2		82
16.13	even	4	inner	16.22.e.a.13.9	yes	82

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.22.e.a.5.9	✓	82	1.1	even	1	trivial
16.22.e.a.13.9	yes	82	16.13	even	4	inner
64.22.e.a.17.2		82	16.3	odd	4
64.22.e.a.49.2		82	4.3	odd	2