Embedded newform 43.9.f.a.22.14

• Label: 43.9.f.a.22.14
• Level: $43$
• Weight: $9$
• Character: 43.22
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $174$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(174\)
Relative dimension:	\(29\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.14
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.911670 + 1.89310i) q^{2} +(5.18580 + 10.7684i) q^{3} +(156.861 + 196.697i) q^{4} +(618.914 - 141.263i) q^{5} -25.1134 q^{6} -3955.82i q^{7} +(-1039.79 + 237.325i) q^{8} +(4001.65 - 5017.91i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.911670	+	1.89310i	−0.0569794	+	0.118319i	−0.927514	−	0.373788i	\(-0.878059\pi\)
							0.870535	+	0.492107i	\(0.163773\pi\)
\(3\)	5.18580	+	10.7684i	0.0640222	+	0.132943i	0.930525	−	0.366230i	\(-0.119351\pi\)
							−0.866502	+	0.499173i	\(0.833637\pi\)
\(5\)	618.914	−	141.263i	0.990263	−	0.226021i	0.303444	−	0.952849i	\(-0.401864\pi\)
							0.686819	+	0.726828i	\(0.259006\pi\)
\(7\)		−	3955.82i		−	1.64757i	−0.566900	−	0.823787i	\(-0.691857\pi\)
							0.566900	−	0.823787i	\(-0.308143\pi\)
\(11\)	−8259.12	+	10356.6i	−0.564109	+	0.707371i	−0.979312	−	0.202358i	\(-0.935139\pi\)
							0.415202	+	0.909729i	\(0.363711\pi\)
\(13\)	409.382	+	1793.62i	0.0143336	+	0.0627997i	0.981589	−	0.191006i	\(-0.0611749\pi\)
							−0.967255	+	0.253805i	\(0.918318\pi\)
\(17\)	28635.5	−	125460.i	0.342854	−	1.50214i	−0.450166	−	0.892945i	\(-0.648635\pi\)
							0.793020	−	0.609195i	\(-0.208507\pi\)
\(19\)	145331.	−	115897.i	1.11517	−	0.889321i	0.120533	−	0.992709i	\(-0.461540\pi\)
							0.994641	+	0.103388i	\(0.0329683\pi\)
\(23\)	48670.1	−	61030.3i	0.173920	−	0.218089i	−0.687229	−	0.726440i	\(-0.741173\pi\)
							0.861150	+	0.508351i	\(0.169745\pi\)
\(29\)	−172907.	+	359045.i	−0.244467	+	0.507641i	−0.986710	−	0.162490i	\(-0.948048\pi\)
							0.742243	+	0.670130i	\(0.233762\pi\)
\(31\)	1.44366e6	+	695230.i	1.56321	+	0.752804i	0.997423	−	0.0717455i	\(-0.0228569\pi\)
							0.565790	+	0.824549i	\(0.308571\pi\)
\(37\)			1.81390e6i			0.967846i	0.875111	+	0.483923i	\(0.160789\pi\)
							−0.875111	+	0.483923i	\(0.839211\pi\)
\(41\)	2.87992e6	+	1.38690e6i	1.01917	+	0.490805i	0.867400	−	0.497612i	\(-0.165790\pi\)
							0.151766	+	0.988416i	\(0.451504\pi\)
\(43\)	−2.37524e6	−	2.45895e6i	−0.694759	−	0.719242i
							\(47\)	173580.	+	217663.i	0.0355721	+	0.0446060i	0.799299	−	0.600933i	\(-0.205204\pi\)
−0.763727	+	0.645539i	\(0.776633\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.14	yes	174
43.2	odd	14	inner	43.9.f.a.2.14	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.14	✓	174	43.2	odd	14	inner
43.9.f.a.22.14	yes	174	1.1	even	1	trivial