Embedded newform 43.9.f.a.22.1

• Label: 43.9.f.a.22.1
• 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			22.1
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.4582 + 27.9461i) q^{2} +(-47.5652 - 98.7701i) q^{3} +(-440.252 - 552.058i) q^{4} +(167.318 - 38.1893i) q^{5} +3400.38 q^{6} +4550.37i q^{7} +(13611.4 - 3106.71i) q^{8} +(-3402.37 + 4266.43i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.4582	+	27.9461i	−0.841135	+	1.74663i	−0.199068	+	0.979986i	\(0.563791\pi\)
							−0.642067	+	0.766648i	\(0.721923\pi\)
\(3\)	−47.5652	−	98.7701i	−0.587224	−	1.21938i	−0.956953	−	0.290244i	\(-0.906264\pi\)
							0.369729	−	0.929140i	\(-0.379451\pi\)
\(5\)	167.318	−	38.1893i	0.267709	−	0.0611028i	−0.0865592	−	0.996247i	\(-0.527587\pi\)
							0.354268	+	0.935144i	\(0.384730\pi\)
\(7\)			4550.37i			1.89520i	0.319463	+	0.947599i	\(0.396497\pi\)
							−0.319463	+	0.947599i	\(0.603503\pi\)
\(11\)	14095.9	−	17675.8i	0.962772	−	1.20728i	−0.0154847	−	0.999880i	\(-0.504929\pi\)
							0.978257	−	0.207398i	\(-0.0664995\pi\)
\(13\)	−2194.69	−	9615.54i	−0.0768420	−	0.336667i	0.921865	−	0.387512i	\(-0.126666\pi\)
							−0.998707	+	0.0508454i	\(0.983808\pi\)
\(17\)	7787.25	−	34118.2i	0.0932370	−	0.408498i	−0.906674	−	0.421832i	\(-0.861387\pi\)
							0.999911	+	0.0133339i	\(0.00424443\pi\)
\(19\)	−21543.4	+	17180.3i	−0.165311	+	0.131831i	−0.702651	−	0.711535i	\(-0.748000\pi\)
							0.537340	+	0.843365i	\(0.319429\pi\)
\(23\)	−32725.8	+	41036.8i	−0.116944	+	0.146643i	−0.836858	−	0.547420i	\(-0.815610\pi\)
							0.719914	+	0.694064i	\(0.244181\pi\)
\(29\)	−355211.	+	737604.i	−0.502221	+	1.04287i	0.483634	+	0.875271i	\(0.339317\pi\)
							−0.985855	+	0.167602i	\(0.946398\pi\)
\(31\)	−1.19506e6	−	575511.i	−1.29403	−	0.623170i	−0.345070	−	0.938577i	\(-0.612145\pi\)
							−0.948957	+	0.315407i	\(0.897859\pi\)
\(37\)			665026.i			0.354839i	0.984135	+	0.177420i	\(0.0567750\pi\)
							−0.984135	+	0.177420i	\(0.943225\pi\)
\(41\)	−102802.	−	49507.0i	−0.0363804	−	0.0175199i	0.415605	−	0.909545i	\(-0.363570\pi\)
							−0.451985	+	0.892025i	\(0.649284\pi\)
\(43\)	3.21441e6	−	1.16439e6i	0.940214	−	0.340583i
							\(47\)	1.40439e6	+	1.76106e6i	0.287805	+	0.360896i	0.904625	−	0.426208i	\(-0.140151\pi\)
−0.616820	+	0.787104i	\(0.711580\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.1	yes	174
43.2	odd	14	inner	43.9.f.a.2.1	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.1	✓	174	43.2	odd	14	inner
43.9.f.a.22.1	yes	174	1.1	even	1	trivial