Embedded newform 168.2.v.a.11.6

• Label: 168.2.v.a.11.6
• Level: $168$
• Weight: $2$
• Character: 168.11
• 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			11.6
Character	\(\chi\)	\(=\)	168.11
Dual form			168.2.v.a.107.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08135 + 0.911422i) q^{2} +(-0.316681 + 1.70285i) q^{3} +(0.338619 - 1.97113i) q^{4} +(1.86088 + 3.22314i) q^{5} +(-1.20958 - 2.13001i) q^{6} +(2.46429 - 0.962962i) q^{7} +(1.43036 + 2.44009i) q^{8} +(-2.79943 - 1.07852i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08135	+	0.911422i	−0.764627	+	0.644473i
							\(3\)	−0.316681	+	1.70285i	−0.182836	+	0.983143i
\(5\)	1.86088	+	3.22314i	0.832210	+	1.44143i								0.896282	+	0.443485i	\(0.146258\pi\)
							−0.0640716	+	0.997945i	\(0.520409\pi\)
\(7\)	2.46429	−	0.962962i	0.931412	−	0.363965i
							\(11\)	−2.26958	−	1.31034i	−0.684304	−	0.395083i	0.117171	−	0.993112i	\(-0.462618\pi\)
−0.801475	+	0.598029i	\(0.795951\pi\)
\(13\)			3.57182i			0.990645i	0.868709	+	0.495323i	\(0.164950\pi\)
							−0.868709	+	0.495323i	\(0.835050\pi\)
\(17\)	0.186192	+	0.107498i	0.0451582	+	0.0260721i	0.522409	−	0.852695i	\(-0.325033\pi\)
							−0.477251	+	0.878767i	\(0.658367\pi\)
\(19\)	−1.14103	−	1.97631i	−0.261769	−	0.453398i	0.704943	−	0.709264i	\(-0.250973\pi\)
							−0.966712	+	0.255866i	\(0.917639\pi\)
\(23\)	−2.33586	−	4.04583i	−0.487061	−	0.843615i	0.512828	−	0.858491i	\(-0.328598\pi\)
							−0.999889	+	0.0148767i	\(0.995264\pi\)
\(29\)	2.57415			0.478008			0.239004	−	0.971019i	\(-0.423179\pi\)
							0.239004	+	0.971019i	\(0.423179\pi\)
\(31\)	4.26196	+	2.46064i	0.765471	+	0.441945i	0.831257	−	0.555889i	\(-0.187622\pi\)
							−0.0657857	+	0.997834i	\(0.520955\pi\)
\(37\)	7.31104	−	4.22103i	1.20193	−	0.693933i	0.240944	−	0.970539i	\(-0.422543\pi\)
							0.960983	+	0.276606i	\(0.0892098\pi\)
\(41\)			0.909442i			0.142031i	0.997475	+	0.0710155i	\(0.0226240\pi\)
							−0.997475	+	0.0710155i	\(0.977376\pi\)
\(43\)	3.73541			0.569644			0.284822	−	0.958580i	\(-0.408065\pi\)
							0.284822	+	0.958580i	\(0.408065\pi\)
\(47\)	−0.586728	−	1.01624i	−0.0855831	−	0.148234i	0.820056	−	0.572283i	\(-0.193942\pi\)
							−0.905640	+	0.424048i	\(0.860609\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.v.a.11.6	yes	56
3.2	odd	2	inner	168.2.v.a.11.23	yes	56
4.3	odd	2		672.2.bd.a.431.16		56
7.2	even	3	inner	168.2.v.a.107.26	yes	56
8.3	odd	2	inner	168.2.v.a.11.3	✓	56
8.5	even	2		672.2.bd.a.431.15		56
12.11	even	2		672.2.bd.a.431.21		56
21.2	odd	6	inner	168.2.v.a.107.3	yes	56
24.5	odd	2		672.2.bd.a.431.22		56
24.11	even	2	inner	168.2.v.a.11.26	yes	56
28.23	odd	6		672.2.bd.a.527.22		56
56.37	even	6		672.2.bd.a.527.21		56
56.51	odd	6	inner	168.2.v.a.107.23	yes	56
84.23	even	6		672.2.bd.a.527.15		56
168.107	even	6	inner	168.2.v.a.107.6	yes	56
168.149	odd	6		672.2.bd.a.527.16		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.3	✓	56	8.3	odd	2	inner
168.2.v.a.11.6	yes	56	1.1	even	1	trivial
168.2.v.a.11.23	yes	56	3.2	odd	2	inner
168.2.v.a.11.26	yes	56	24.11	even	2	inner
168.2.v.a.107.3	yes	56	21.2	odd	6	inner
168.2.v.a.107.6	yes	56	168.107	even	6	inner
168.2.v.a.107.23	yes	56	56.51	odd	6	inner
168.2.v.a.107.26	yes	56	7.2	even	3	inner
672.2.bd.a.431.15		56	8.5	even	2
672.2.bd.a.431.16		56	4.3	odd	2
672.2.bd.a.431.21		56	12.11	even	2
672.2.bd.a.431.22		56	24.5	odd	2
672.2.bd.a.527.15		56	84.23	even	6
672.2.bd.a.527.16		56	168.149	odd	6
672.2.bd.a.527.21		56	56.37	even	6
672.2.bd.a.527.22		56	28.23	odd	6