Newform orbit 168.2.bc.a

• Label: 168.2.bc.a
• Level: $168$
• Weight: $2$
• Character orbit: 168.bc
• Analytic conductor: $1.341$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34148675396\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 2 q^{4} - 16 q^{8} + 16 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−1.40107	−	0.192386i	0.866025	−	0.500000i	1.92598	+	0.539091i	−2.93503	−	1.69454i	−1.30955	+	0.533922i	−1.85242	−	1.88906i	−2.59471	−	1.12583i	0.500000	−	0.866025i	3.78617	+	2.93882i
37.2	−1.33630	+	0.462932i	−0.866025	+	0.500000i	1.57139	−	1.23723i	−1.56250	−	0.902108i	0.925802	−	1.06906i	2.63683	−	0.217074i	−1.52709	+	2.38076i	0.500000	−	0.866025i	2.50558	+	0.482155i
37.3	−1.31787	−	0.513056i	−0.866025	+	0.500000i	1.47355	+	1.35228i	−0.0402223	−	0.0232224i	1.39783	−	0.214614i	−1.97032	+	1.76574i	−1.24814	−	2.53814i	0.500000	−	0.866025i	0.0410933	+	0.0512403i
37.4	−1.00926	−	0.990649i	0.866025	−	0.500000i	0.0372299	+	1.99965i	0.586448	+	0.338586i	−1.36937	−	0.353295i	2.23683	+	1.41301i	1.94338	−	2.05506i	0.500000	−	0.866025i	−0.256462	−	0.922687i
37.5	−0.938973	+	1.05751i	0.866025	−	0.500000i	−0.236659	−	1.98595i	1.98722	+	1.14732i	−0.284419	+	1.38532i	−1.05630	−	2.42574i	2.32238	+	1.61448i	0.500000	−	0.866025i	−3.07926	+	1.02420i
37.6	−0.446345	+	1.34193i	−0.866025	+	0.500000i	−1.60155	−	1.19793i	−1.98722	−	1.14732i	−0.284419	−	1.38532i	−1.05630	−	2.42574i	2.32238	−	1.61448i	0.500000	−	0.866025i	2.42662	−	2.15461i
37.7	−0.189716	−	1.40143i	−0.866025	+	0.500000i	−1.92802	+	0.531748i	3.09843	+	1.78888i	0.865014	+	1.11882i	0.993295	+	2.45222i	1.11098	+	2.60110i	0.500000	−	0.866025i	1.91917	−	4.68162i
37.8	0.267238	+	1.38873i	0.866025	−	0.500000i	−1.85717	+	0.742246i	1.56250	+	0.902108i	0.925802	+	1.06906i	2.63683	−	0.217074i	−1.52709	−	2.38076i	0.500000	−	0.866025i	−0.835229	+	2.41097i
37.9	0.268038	−	1.38858i	0.866025	−	0.500000i	−1.85631	−	0.744384i	−1.23074	−	0.710569i	−0.462163	−	1.33656i	1.39545	−	2.24783i	−1.53120	+	2.37811i	0.500000	−	0.866025i	−1.31657	+	1.51852i
37.10	0.491996	−	1.32587i	−0.866025	+	0.500000i	−1.51588	−	1.30465i	−3.08781	−	1.78275i	0.236856	+	1.39424i	−2.38336	+	1.14873i	−2.47560	+	1.36799i	0.500000	−	0.866025i	−3.88289	+	3.21694i
37.11	0.867144	+	1.11717i	−0.866025	+	0.500000i	−0.496121	+	1.93749i	2.93503	+	1.69454i	−1.30955	−	0.533922i	−1.85242	−	1.88906i	−2.59471	+	1.12583i	0.500000	−	0.866025i	0.652012	+	4.74833i
37.12	0.902242	−	1.08902i	0.866025	−	0.500000i	−0.371918	−	1.96512i	3.08781	+	1.78275i	0.236856	−	1.39424i	−2.38336	+	1.14873i	−2.47560	−	1.36799i	0.500000	−	0.866025i	4.72740	−	1.75421i
37.13	1.06853	−	0.926418i	−0.866025	+	0.500000i	0.283500	−	1.97980i	1.23074	+	0.710569i	−0.462163	+	1.33656i	1.39545	−	2.24783i	−1.53120	−	2.37811i	0.500000	−	0.866025i	1.97336	−	0.380919i
37.14	1.10325	+	0.884778i	0.866025	−	0.500000i	0.434335	+	1.95227i	0.0402223	+	0.0232224i	1.39783	+	0.214614i	−1.97032	+	1.76574i	−1.24814	+	2.53814i	0.500000	−	0.866025i	0.0238287	+	0.0612079i
37.15	1.30853	−	0.536416i	0.866025	−	0.500000i	1.42451	−	1.40384i	−3.09843	−	1.78888i	0.865014	−	1.11882i	0.993295	+	2.45222i	1.11098	−	2.60110i	0.500000	−	0.866025i	−5.01398	−	0.678758i
37.16	1.36256	+	0.378724i	−0.866025	+	0.500000i	1.71314	+	1.03207i	−0.586448	−	0.338586i	−1.36937	+	0.353295i	2.23683	+	1.41301i	1.94338	+	2.05506i	0.500000	−	0.866025i	−0.670840	−	0.683446i
109.1	−1.40107	+	0.192386i	0.866025	+	0.500000i	1.92598	−	0.539091i	−2.93503	+	1.69454i	−1.30955	−	0.533922i	−1.85242	+	1.88906i	−2.59471	+	1.12583i	0.500000	+	0.866025i	3.78617	−	2.93882i
109.2	−1.33630	−	0.462932i	−0.866025	−	0.500000i	1.57139	+	1.23723i	−1.56250	+	0.902108i	0.925802	+	1.06906i	2.63683	+	0.217074i	−1.52709	−	2.38076i	0.500000	+	0.866025i	2.50558	−	0.482155i
109.3	−1.31787	+	0.513056i	−0.866025	−	0.500000i	1.47355	−	1.35228i	−0.0402223	+	0.0232224i	1.39783	+	0.214614i	−1.97032	−	1.76574i	−1.24814	+	2.53814i	0.500000	+	0.866025i	0.0410933	−	0.0512403i
109.4	−1.00926	+	0.990649i	0.866025	+	0.500000i	0.0372299	−	1.99965i	0.586448	−	0.338586i	−1.36937	+	0.353295i	2.23683	−	1.41301i	1.94338	+	2.05506i	0.500000	+	0.866025i	−0.256462	+	0.922687i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
8.b	even	2	1	inner
56.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	168.2.bc.a	✓	32
3.b	odd	2	1		504.2.cj.e		32
4.b	odd	2	1		672.2.bk.a		32
7.c	even	3	1	inner	168.2.bc.a	✓	32
7.c	even	3	1		1176.2.c.e		16
7.d	odd	6	1		1176.2.c.f		16
8.b	even	2	1	inner	168.2.bc.a	✓	32
8.d	odd	2	1		672.2.bk.a		32
12.b	even	2	1		2016.2.cr.e		32
21.h	odd	6	1		504.2.cj.e		32
24.f	even	2	1		2016.2.cr.e		32
24.h	odd	2	1		504.2.cj.e		32
28.f	even	6	1		4704.2.c.f		16
28.g	odd	6	1		672.2.bk.a		32
28.g	odd	6	1		4704.2.c.e		16
56.j	odd	6	1		1176.2.c.f		16
56.k	odd	6	1		672.2.bk.a		32
56.k	odd	6	1		4704.2.c.e		16
56.m	even	6	1		4704.2.c.f		16
56.p	even	6	1	inner	168.2.bc.a	✓	32
56.p	even	6	1		1176.2.c.e		16
84.n	even	6	1		2016.2.cr.e		32
168.s	odd	6	1		504.2.cj.e		32
168.v	even	6	1		2016.2.cr.e		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.2.bc.a	✓	32	1.a	even	1	1	trivial
168.2.bc.a	✓	32	7.c	even	3	1	inner
168.2.bc.a	✓	32	8.b	even	2	1	inner
168.2.bc.a	✓	32	56.p	even	6	1	inner
504.2.cj.e		32	3.b	odd	2	1
504.2.cj.e		32	21.h	odd	6	1
504.2.cj.e		32	24.h	odd	2	1
504.2.cj.e		32	168.s	odd	6	1
672.2.bk.a		32	4.b	odd	2	1
672.2.bk.a		32	8.d	odd	2	1
672.2.bk.a		32	28.g	odd	6	1
672.2.bk.a		32	56.k	odd	6	1
1176.2.c.e		16	7.c	even	3	1
1176.2.c.e		16	56.p	even	6	1
1176.2.c.f		16	7.d	odd	6	1
1176.2.c.f		16	56.j	odd	6	1
2016.2.cr.e		32	12.b	even	2	1
2016.2.cr.e		32	24.f	even	2	1
2016.2.cr.e		32	84.n	even	6	1
2016.2.cr.e		32	168.v	even	6	1
4704.2.c.e		16	28.g	odd	6	1
4704.2.c.e		16	56.k	odd	6	1
4704.2.c.f		16	28.f	even	6	1
4704.2.c.f		16	56.m	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(168, [\chi])\).