Embedded newform 168.2.v.a.11.4

• Label: 168.2.v.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	168.11
Dual form			168.2.v.a.107.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32990 - 0.481016i) q^{2} +(1.28463 + 1.16178i) q^{3} +(1.53725 + 1.27940i) q^{4} +(-0.646402 - 1.11960i) q^{5} +(-1.14959 - 2.16297i) q^{6} +(2.42742 - 1.05244i) q^{7} +(-1.42896 - 2.44091i) q^{8} +(0.300539 + 2.98491i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32990	−	0.481016i	−0.940378	−	0.340130i
							\(3\)	1.28463	+	1.16178i	0.741680	+	0.670753i
\(5\)	−0.646402	−	1.11960i	−0.289080	−	0.500700i								0.684511	−	0.729003i	\(-0.260016\pi\)
							−0.973590	+	0.228302i	\(0.926683\pi\)
\(7\)	2.42742	−	1.05244i	0.917478	−	0.397786i
							\(11\)	1.60004	+	0.923782i	0.482430	+	0.278531i	0.721428	−	0.692489i	\(-0.243486\pi\)
−0.238999	+	0.971020i	\(0.576819\pi\)
\(13\)			2.25432i			0.625235i	0.949879	+	0.312617i	\(0.101206\pi\)
							−0.949879	+	0.312617i	\(0.898794\pi\)
\(17\)	3.89580	+	2.24924i	0.944871	+	0.545522i	0.891484	−	0.453052i	\(-0.149665\pi\)
							0.0533874	+	0.998574i	\(0.482998\pi\)
\(19\)	−2.80851	−	4.86448i	−0.644317	−	1.11599i	−0.984459	−	0.175615i	\(-0.943809\pi\)
							0.340142	−	0.940374i	\(-0.389525\pi\)
\(23\)	0.519880	+	0.900459i	0.108402	+	0.187759i	0.915123	−	0.403174i	\(-0.132093\pi\)
							−0.806721	+	0.590933i	\(0.798760\pi\)
\(29\)	−1.32085			−0.245277			−0.122638	−	0.992451i	\(-0.539135\pi\)
							−0.122638	+	0.992451i	\(0.539135\pi\)
\(31\)	−3.69700	−	2.13446i	−0.664000	−	0.383361i	0.129799	−	0.991540i	\(-0.458567\pi\)
							−0.793800	+	0.608179i	\(0.791900\pi\)
\(37\)	−8.18394	+	4.72500i	−1.34543	+	0.776786i	−0.987599	−	0.157000i	\(-0.949818\pi\)
							−0.357834	+	0.933785i	\(0.616484\pi\)
\(41\)		−	1.39634i		−	0.218072i	−0.994038	−	0.109036i	\(-0.965224\pi\)
							0.994038	−	0.109036i	\(-0.0347764\pi\)
\(43\)	−6.02578			−0.918922			−0.459461	−	0.888198i	\(-0.651957\pi\)
							−0.459461	+	0.888198i	\(0.651957\pi\)
\(47\)	−5.90249	−	10.2234i	−0.860966	−	1.49124i	−0.870997	−	0.491288i	\(-0.836526\pi\)
							0.0100312	−	0.999950i	\(-0.496807\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.v.a.11.4	✓	56
3.2	odd	2	inner	168.2.v.a.11.25	yes	56
4.3	odd	2		672.2.bd.a.431.5		56
7.2	even	3	inner	168.2.v.a.107.16	yes	56
8.3	odd	2	inner	168.2.v.a.11.13	yes	56
8.5	even	2		672.2.bd.a.431.6		56
12.11	even	2		672.2.bd.a.431.26		56
21.2	odd	6	inner	168.2.v.a.107.13	yes	56
24.5	odd	2		672.2.bd.a.431.25		56
24.11	even	2	inner	168.2.v.a.11.16	yes	56
28.23	odd	6		672.2.bd.a.527.25		56
56.37	even	6		672.2.bd.a.527.26		56
56.51	odd	6	inner	168.2.v.a.107.25	yes	56
84.23	even	6		672.2.bd.a.527.6		56
168.107	even	6	inner	168.2.v.a.107.4	yes	56
168.149	odd	6		672.2.bd.a.527.5		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.4	✓	56	1.1	even	1	trivial
168.2.v.a.11.13	yes	56	8.3	odd	2	inner
168.2.v.a.11.16	yes	56	24.11	even	2	inner
168.2.v.a.11.25	yes	56	3.2	odd	2	inner
168.2.v.a.107.4	yes	56	168.107	even	6	inner
168.2.v.a.107.13	yes	56	21.2	odd	6	inner
168.2.v.a.107.16	yes	56	7.2	even	3	inner
168.2.v.a.107.25	yes	56	56.51	odd	6	inner
672.2.bd.a.431.5		56	4.3	odd	2
672.2.bd.a.431.6		56	8.5	even	2
672.2.bd.a.431.25		56	24.5	odd	2
672.2.bd.a.431.26		56	12.11	even	2
672.2.bd.a.527.5		56	168.149	odd	6
672.2.bd.a.527.6		56	84.23	even	6
672.2.bd.a.527.25		56	28.23	odd	6
672.2.bd.a.527.26		56	56.37	even	6