Embedded newform 168.2.v.a.107.7

• Label: 168.2.v.a.107.7
• 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			107.7
Character	\(\chi\)	\(=\)	168.107
Dual form			168.2.v.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.982844 + 1.01687i) q^{2} +(-0.987527 + 1.42295i) q^{3} +(-0.0680365 - 1.99884i) q^{4} +(1.77783 - 3.07929i) q^{5} +(-0.476367 - 2.40272i) q^{6} +(0.793456 - 2.52397i) q^{7} +(2.09943 + 1.89537i) q^{8} +(-1.04958 - 2.81041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.982844	+	1.01687i	−0.694975	+	0.719033i
							\(3\)	−0.987527	+	1.42295i	−0.570149	+	0.821541i
\(5\)	1.77783	−	3.07929i	0.795070	−	1.37710i								−0.127724	−	0.991810i	\(-0.540767\pi\)
							0.922795	−	0.385292i	\(-0.125899\pi\)
\(7\)	0.793456	−	2.52397i	0.299898	−	0.953971i
							\(11\)	−0.396703	+	0.229036i	−0.119610	+	0.0690571i	−0.558612	−	0.829429i	\(-0.688666\pi\)
0.439001	+	0.898486i	\(0.355332\pi\)
\(13\)			0.799063i			0.221620i	0.993842	+	0.110810i	\(0.0353445\pi\)
							−0.993842	+	0.110810i	\(0.964655\pi\)
\(17\)	5.48456	−	3.16651i	1.33020	−	0.767993i	0.344872	−	0.938650i	\(-0.387922\pi\)
							0.985330	+	0.170657i	\(0.0545891\pi\)
\(19\)	−2.61696	+	4.53272i	−0.600373	+	1.03988i	0.392392	+	0.919798i	\(0.371648\pi\)
							−0.992764	+	0.120078i	\(0.961686\pi\)
\(23\)	1.55053	−	2.68559i	0.323307	−	0.559985i	−0.657861	−	0.753139i	\(-0.728539\pi\)
							0.981168	+	0.193155i	\(0.0618719\pi\)
\(29\)	−2.60340			−0.483440			−0.241720	−	0.970346i	\(-0.577712\pi\)
							−0.241720	+	0.970346i	\(0.577712\pi\)
\(31\)	4.42520	−	2.55489i	0.794790	−	0.458872i	−0.0468563	−	0.998902i	\(-0.514920\pi\)
							0.841646	+	0.540030i	\(0.181587\pi\)
\(37\)	−2.89424	−	1.67099i	−0.475810	−	0.274709i	0.242859	−	0.970062i	\(-0.421915\pi\)
							−0.718669	+	0.695352i	\(0.755248\pi\)
\(41\)			10.1452i			1.58441i	0.610256	+	0.792205i	\(0.291067\pi\)
							−0.610256	+	0.792205i	\(0.708933\pi\)
\(43\)	3.56359			0.543442			0.271721	−	0.962376i	\(-0.412407\pi\)
							0.271721	+	0.962376i	\(0.412407\pi\)
\(47\)	−2.15923	+	3.73989i	−0.314956	+	0.545519i	−0.979428	−	0.201794i	\(-0.935323\pi\)
							0.664472	+	0.747313i	\(0.268656\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.v.a.107.7	yes	56
3.2	odd	2	inner	168.2.v.a.107.22	yes	56
4.3	odd	2		672.2.bd.a.527.20		56
7.4	even	3	inner	168.2.v.a.11.12	yes	56
8.3	odd	2	inner	168.2.v.a.107.17	yes	56
8.5	even	2		672.2.bd.a.527.19		56
12.11	even	2		672.2.bd.a.527.1		56
21.11	odd	6	inner	168.2.v.a.11.17	yes	56
24.5	odd	2		672.2.bd.a.527.2		56
24.11	even	2	inner	168.2.v.a.107.12	yes	56
28.11	odd	6		672.2.bd.a.431.2		56
56.11	odd	6	inner	168.2.v.a.11.22	yes	56
56.53	even	6		672.2.bd.a.431.1		56
84.11	even	6		672.2.bd.a.431.19		56
168.11	even	6	inner	168.2.v.a.11.7	✓	56
168.53	odd	6		672.2.bd.a.431.20		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.7	✓	56	168.11	even	6	inner
168.2.v.a.11.12	yes	56	7.4	even	3	inner
168.2.v.a.11.17	yes	56	21.11	odd	6	inner
168.2.v.a.11.22	yes	56	56.11	odd	6	inner
168.2.v.a.107.7	yes	56	1.1	even	1	trivial
168.2.v.a.107.12	yes	56	24.11	even	2	inner
168.2.v.a.107.17	yes	56	8.3	odd	2	inner
168.2.v.a.107.22	yes	56	3.2	odd	2	inner
672.2.bd.a.431.1		56	56.53	even	6
672.2.bd.a.431.2		56	28.11	odd	6
672.2.bd.a.431.19		56	84.11	even	6
672.2.bd.a.431.20		56	168.53	odd	6
672.2.bd.a.527.1		56	12.11	even	2
672.2.bd.a.527.2		56	24.5	odd	2
672.2.bd.a.527.19		56	8.5	even	2
672.2.bd.a.527.20		56	4.3	odd	2