Embedded newform 161.4.g.a.45.20

• Label: 161.4.g.a.45.20
• Level: $161$
• Weight: $4$
• Character: 161.45
• Analytic conductor: $9.499$
• Analytic rank: $0$
• Dimension: $92$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	161.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.49930751092\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(46\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			45.20
Character	\(\chi\)	\(=\)	161.45
Dual form			161.4.g.a.68.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.475042 - 0.822798i) q^{2} +(-3.34199 - 1.92950i) q^{3} +(3.54867 - 6.14648i) q^{4} +(8.95281 + 15.5067i) q^{5} +3.66637i q^{6} +(-3.46541 + 18.1932i) q^{7} -14.3438 q^{8} +(-6.05407 - 10.4860i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 6 q^{3} - 168 q^{4} + 60 q^{8} + 348 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).

\(n\)	\(24\)	\(120\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.475042	−	0.822798i	−0.167953	−	0.290903i	0.769747	−	0.638349i	\(-0.220382\pi\)
							−0.937700	+	0.347446i	\(0.887049\pi\)
\(3\)	−3.34199	−	1.92950i	−0.643166	−	0.371332i	0.142667	−	0.989771i	\(-0.454432\pi\)
							−0.785833	+	0.618439i	\(0.787766\pi\)
\(5\)	8.95281	+	15.5067i	0.800763	+	1.38696i	0.919114	+	0.393991i	\(0.128906\pi\)
							−0.118351	+	0.992972i	\(0.537761\pi\)
\(7\)	−3.46541	+	18.1932i	−0.187115	+	0.982338i
							\(11\)	−40.8764	−	23.6000i	−1.12043	−	0.646879i	−0.178917	−	0.983864i	\(-0.557259\pi\)
−0.941510	+	0.336985i	\(0.890593\pi\)
\(13\)			70.3388i			1.50065i	0.661068	+	0.750326i	\(0.270103\pi\)
							−0.661068	+	0.750326i	\(0.729897\pi\)
\(17\)	−57.2873	+	99.2246i	−0.817307	+	1.41562i	0.0903521	+	0.995910i	\(0.471201\pi\)
							−0.907659	+	0.419708i	\(0.862133\pi\)
\(19\)	26.1852	+	45.3540i	0.316173	+	0.547628i	0.979686	−	0.200537i	\(-0.0642686\pi\)
							−0.663513	+	0.748165i	\(0.730935\pi\)
\(23\)	109.997	+	8.22288i	0.997217	+	0.0745474i
							\(29\)	266.217			1.70466			0.852332	−	0.523001i	\(-0.175188\pi\)
0.852332	+	0.523001i	\(0.175188\pi\)
\(31\)	−221.724	−	128.013i	−1.28461	−	0.741669i	−0.306921	−	0.951735i	\(-0.599299\pi\)
							−0.977687	+	0.210066i	\(0.932632\pi\)
\(37\)	−107.438	+	62.0294i	−0.477371	+	0.275610i	−0.719320	−	0.694679i	\(-0.755547\pi\)
							0.241949	+	0.970289i	\(0.422213\pi\)
\(41\)			94.4673i			0.359837i	0.983682	+	0.179918i	\(0.0575834\pi\)
							−0.983682	+	0.179918i	\(0.942417\pi\)
\(43\)			95.3018i			0.337986i	0.985617	+	0.168993i	\(0.0540515\pi\)
							−0.985617	+	0.168993i	\(0.945948\pi\)
\(47\)	149.392	−	86.2513i	0.463639	−	0.267682i	−0.249934	−	0.968263i	\(-0.580409\pi\)
							0.713573	+	0.700581i	\(0.247076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.4.g.a.45.20	yes	92
7.5	odd	6	inner	161.4.g.a.68.19	yes	92
23.22	odd	2	inner	161.4.g.a.45.19	✓	92
161.68	even	6	inner	161.4.g.a.68.20	yes	92

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.4.g.a.45.19	✓	92	23.22	odd	2	inner
161.4.g.a.45.20	yes	92	1.1	even	1	trivial
161.4.g.a.68.19	yes	92	7.5	odd	6	inner
161.4.g.a.68.20	yes	92	161.68	even	6	inner