Embedded newform 168.2.v.a.107.2

• Label: 168.2.v.a.107.2
• Level: $168$
• Weight: $2$
• Character: 168.107
• Analytic conductor: $1.341$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34148675396\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			107.2
Character	\(\chi\)	\(=\)	168.107
Dual form			168.2.v.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40124 - 0.191140i) q^{2} +(1.66017 + 0.493786i) q^{3} +(1.92693 + 0.535666i) q^{4} +(1.09212 - 1.89161i) q^{5} +(-2.23191 - 1.00924i) q^{6} +(-0.451847 - 2.60688i) q^{7} +(-2.59770 - 1.11891i) q^{8} +(2.51235 + 1.63954i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40124	−	0.191140i	−0.990824	−	0.135157i
							\(3\)	1.66017	+	0.493786i	0.958502	+	0.285087i
\(5\)	1.09212	−	1.89161i	0.488411	−	0.845953i								−0.511500	−	0.859283i	\(-0.670910\pi\)
							0.999911	+	0.0133302i	\(0.00424327\pi\)
\(7\)	−0.451847	−	2.60688i	−0.170782	−	0.985309i
							\(11\)	−1.45052	+	0.837460i	−0.437349	+	0.252504i	−0.702473	−	0.711711i	\(-0.747921\pi\)
0.265123	+	0.964215i	\(0.414587\pi\)
\(13\)			1.56085i			0.432903i	0.976293	+	0.216452i	\(0.0694483\pi\)
							−0.976293	+	0.216452i	\(0.930552\pi\)
\(17\)	−0.278795	+	0.160963i	−0.0676178	+	0.0390392i	−0.533428	−	0.845846i	\(-0.679096\pi\)
							0.465810	+	0.884885i	\(0.345763\pi\)
\(19\)	2.87437	−	4.97856i	0.659427	−	1.14216i	−0.321337	−	0.946965i	\(-0.604132\pi\)
							0.980764	−	0.195196i	\(-0.0625343\pi\)
\(23\)	−3.26717	+	5.65891i	−0.681252	+	1.17996i	0.293346	+	0.956006i	\(0.405231\pi\)
							−0.974599	+	0.223958i	\(0.928102\pi\)
\(29\)	7.04205			1.30768			0.653838	−	0.756635i	\(-0.273158\pi\)
							0.653838	+	0.756635i	\(0.273158\pi\)
\(31\)	−7.76983	+	4.48591i	−1.39550	+	0.805694i	−0.993917	−	0.110128i	\(-0.964874\pi\)
							−0.401585	+	0.915822i	\(0.631541\pi\)
\(37\)	0.946173	+	0.546273i	0.155550	+	0.0898068i	0.575755	−	0.817623i	\(-0.304708\pi\)
							−0.420205	+	0.907429i	\(0.638042\pi\)
\(41\)			3.44933i			0.538694i	0.963043	+	0.269347i	\(0.0868079\pi\)
							−0.963043	+	0.269347i	\(0.913192\pi\)
\(43\)	−11.6570			−1.77768			−0.888841	−	0.458215i	\(-0.848489\pi\)
							−0.888841	+	0.458215i	\(0.848489\pi\)
\(47\)	−2.25371	+	3.90354i	−0.328737	+	0.569390i	−0.982262	−	0.187516i	\(-0.939956\pi\)
							0.653524	+	0.756906i	\(0.273290\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.v.a.107.2	yes	56
3.2	odd	2	inner	168.2.v.a.107.27	yes	56
4.3	odd	2		672.2.bd.a.527.4		56
7.4	even	3	inner	168.2.v.a.11.20	yes	56
8.3	odd	2	inner	168.2.v.a.107.9	yes	56
8.5	even	2		672.2.bd.a.527.3		56
12.11	even	2		672.2.bd.a.527.17		56
21.11	odd	6	inner	168.2.v.a.11.9	yes	56
24.5	odd	2		672.2.bd.a.527.18		56
24.11	even	2	inner	168.2.v.a.107.20	yes	56
28.11	odd	6		672.2.bd.a.431.18		56
56.11	odd	6	inner	168.2.v.a.11.27	yes	56
56.53	even	6		672.2.bd.a.431.17		56
84.11	even	6		672.2.bd.a.431.3		56
168.11	even	6	inner	168.2.v.a.11.2	✓	56
168.53	odd	6		672.2.bd.a.431.4		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.2	✓	56	168.11	even	6	inner
168.2.v.a.11.9	yes	56	21.11	odd	6	inner
168.2.v.a.11.20	yes	56	7.4	even	3	inner
168.2.v.a.11.27	yes	56	56.11	odd	6	inner
168.2.v.a.107.2	yes	56	1.1	even	1	trivial
168.2.v.a.107.9	yes	56	8.3	odd	2	inner
168.2.v.a.107.20	yes	56	24.11	even	2	inner
168.2.v.a.107.27	yes	56	3.2	odd	2	inner
672.2.bd.a.431.3		56	84.11	even	6
672.2.bd.a.431.4		56	168.53	odd	6
672.2.bd.a.431.17		56	56.53	even	6
672.2.bd.a.431.18		56	28.11	odd	6
672.2.bd.a.527.3		56	8.5	even	2
672.2.bd.a.527.4		56	4.3	odd	2
672.2.bd.a.527.17		56	12.11	even	2
672.2.bd.a.527.18		56	24.5	odd	2