Embedded newform 224.3.w.a.43.17

• Label: 224.3.w.a.43.17
• Level: $224$
• Weight: $3$
• Character: 224.43
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			43.17
Character	\(\chi\)	\(=\)	224.43
Dual form			224.3.w.a.99.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.703873 - 1.87205i) q^{2} +(0.0651792 + 0.157357i) q^{3} +(-3.00913 + 2.63537i) q^{4} +(3.56366 - 8.60344i) q^{5} +(0.248701 - 0.232778i) q^{6} +(-1.87083 + 1.87083i) q^{7} +(7.05157 + 3.77827i) q^{8} +(6.34345 - 6.34345i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.703873	−	1.87205i	−0.351936	−	0.936024i
							\(3\)	0.0651792	+	0.157357i	0.0217264	+	0.0524522i	0.934370	−	0.356303i	\(-0.115963\pi\)
−0.912644	+	0.408755i	\(0.865963\pi\)
\(5\)	3.56366	−	8.60344i	0.712732	−	1.72069i	0.0196740	−	0.999806i	\(-0.493737\pi\)
							0.693058	−	0.720882i	\(-0.256263\pi\)
\(7\)	−1.87083	+	1.87083i	−0.267261	+	0.267261i
							\(11\)	0.0382251	−	0.0922835i	0.00347501	−	0.00838941i	−0.922133	−	0.386874i	\(-0.873555\pi\)
0.925608	+	0.378485i	\(0.123555\pi\)
\(13\)	−0.154435	−	0.372838i	−0.0118796	−	0.0286799i	0.917828	−	0.396977i	\(-0.129941\pi\)
							−0.929708	+	0.368297i	\(0.879941\pi\)
\(17\)		−	8.50235i		−	0.500138i	−0.968228	−	0.250069i	\(-0.919547\pi\)
							0.968228	−	0.250069i	\(-0.0804533\pi\)
\(19\)	−24.2726	+	10.0540i	−1.27750	+	0.529160i	−0.915238	−	0.402915i	\(-0.867997\pi\)
							−0.362267	+	0.932074i	\(0.617997\pi\)
\(23\)	−17.8463	−	17.8463i	−0.775926	−	0.775926i	0.203209	−	0.979135i	\(-0.434863\pi\)
							−0.979135	+	0.203209i	\(0.934863\pi\)
\(29\)	32.5318	−	13.4751i	1.12179	−	0.464659i	0.256805	−	0.966463i	\(-0.417330\pi\)
							0.864981	+	0.501805i	\(0.167330\pi\)
\(31\)		−	9.33478i		−	0.301122i	−0.988601	−	0.150561i	\(-0.951892\pi\)
							0.988601	−	0.150561i	\(-0.0481080\pi\)
\(37\)	−14.2220	+	34.3349i	−0.384378	+	0.927970i	0.606730	+	0.794908i	\(0.292481\pi\)
							−0.991108	+	0.133062i	\(0.957519\pi\)
\(41\)	−36.4939	+	36.4939i	−0.890094	+	0.890094i	−0.994531	−	0.104437i	\(-0.966696\pi\)
							0.104437	+	0.994531i	\(0.466696\pi\)
\(43\)	10.0412	−	24.2416i	0.233516	−	0.563758i	−0.763070	−	0.646316i	\(-0.776309\pi\)
							0.996586	+	0.0825579i	\(0.0263089\pi\)
\(47\)	45.0631			0.958789			0.479395	−	0.877599i	\(-0.340856\pi\)
							0.479395	+	0.877599i	\(0.340856\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.w.a.43.17	✓	192
32.3	odd	8	inner	224.3.w.a.99.17	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.w.a.43.17	✓	192	1.1	even	1	trivial
224.3.w.a.99.17	yes	192	32.3	odd	8	inner