Newform orbit 504.2.cj.e

• Label: 504.2.cj.e
• Level: $504$
• Weight: $2$
• Character orbit: 504.cj
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.cj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

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

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	−1.36256	−	0.378724i		0		1.71314	+	1.03207i	0.586448	+	0.338586i		0		2.23683	+	1.41301i	−1.94338	−	2.05506i		0		−0.670840	−	0.683446i
37.2	−1.30853	+	0.536416i		0		1.42451	−	1.40384i	3.09843	+	1.78888i		0		0.993295	+	2.45222i	−1.11098	+	2.60110i		0		−5.01398	−	0.678758i
37.3	−1.10325	−	0.884778i		0		0.434335	+	1.95227i	−0.0402223	−	0.0232224i		0		−1.97032	+	1.76574i	1.24814	−	2.53814i		0		0.0238287	+	0.0612079i
37.4	−1.06853	+	0.926418i		0		0.283500	−	1.97980i	−1.23074	−	0.710569i		0		1.39545	−	2.24783i	1.53120	+	2.37811i		0		1.97336	−	0.380919i
37.5	−0.902242	+	1.08902i		0		−0.371918	−	1.96512i	−3.08781	−	1.78275i		0		−2.38336	+	1.14873i	2.47560	+	1.36799i		0		4.72740	−	1.75421i
37.6	−0.867144	−	1.11717i		0		−0.496121	+	1.93749i	−2.93503	−	1.69454i		0		−1.85242	−	1.88906i	2.59471	−	1.12583i		0		0.652012	+	4.74833i
37.7	−0.491996	+	1.32587i		0		−1.51588	−	1.30465i	3.08781	+	1.78275i		0		−2.38336	+	1.14873i	2.47560	−	1.36799i		0		−3.88289	+	3.21694i
37.8	−0.268038	+	1.38858i		0		−1.85631	−	0.744384i	1.23074	+	0.710569i		0		1.39545	−	2.24783i	1.53120	−	2.37811i		0		−1.31657	+	1.51852i
37.9	−0.267238	−	1.38873i		0		−1.85717	+	0.742246i	−1.56250	−	0.902108i		0		2.63683	−	0.217074i	1.52709	+	2.38076i		0		−0.835229	+	2.41097i
37.10	0.189716	+	1.40143i		0		−1.92802	+	0.531748i	−3.09843	−	1.78888i		0		0.993295	+	2.45222i	−1.11098	−	2.60110i		0		1.91917	−	4.68162i
37.11	0.446345	−	1.34193i		0		−1.60155	−	1.19793i	1.98722	+	1.14732i		0		−1.05630	−	2.42574i	−2.32238	+	1.61448i		0		2.42662	−	2.15461i
37.12	0.938973	−	1.05751i		0		−0.236659	−	1.98595i	−1.98722	−	1.14732i		0		−1.05630	−	2.42574i	−2.32238	−	1.61448i		0		−3.07926	+	1.02420i
37.13	1.00926	+	0.990649i		0		0.0372299	+	1.99965i	−0.586448	−	0.338586i		0		2.23683	+	1.41301i	−1.94338	+	2.05506i		0		−0.256462	−	0.922687i
37.14	1.31787	+	0.513056i		0		1.47355	+	1.35228i	0.0402223	+	0.0232224i		0		−1.97032	+	1.76574i	1.24814	+	2.53814i		0		0.0410933	+	0.0512403i
37.15	1.33630	−	0.462932i		0		1.57139	−	1.23723i	1.56250	+	0.902108i		0		2.63683	−	0.217074i	1.52709	−	2.38076i		0		2.50558	+	0.482155i
37.16	1.40107	+	0.192386i		0		1.92598	+	0.539091i	2.93503	+	1.69454i		0		−1.85242	−	1.88906i	2.59471	+	1.12583i		0		3.78617	+	2.93882i
109.1	−1.36256	+	0.378724i		0		1.71314	−	1.03207i	0.586448	−	0.338586i		0		2.23683	−	1.41301i	−1.94338	+	2.05506i		0		−0.670840	+	0.683446i
109.2	−1.30853	−	0.536416i		0		1.42451	+	1.40384i	3.09843	−	1.78888i		0		0.993295	−	2.45222i	−1.11098	−	2.60110i		0		−5.01398	+	0.678758i
109.3	−1.10325	+	0.884778i		0		0.434335	−	1.95227i	−0.0402223	+	0.0232224i		0		−1.97032	−	1.76574i	1.24814	+	2.53814i		0		0.0238287	−	0.0612079i
109.4	−1.06853	−	0.926418i		0		0.283500	+	1.97980i	−1.23074	+	0.710569i		0		1.39545	+	2.24783i	1.53120	−	2.37811i		0		1.97336	+	0.380919i

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	504.2.cj.e		32
3.b	odd	2	1		168.2.bc.a	✓	32
4.b	odd	2	1		2016.2.cr.e		32
7.c	even	3	1	inner	504.2.cj.e		32
8.b	even	2	1	inner	504.2.cj.e		32
8.d	odd	2	1		2016.2.cr.e		32
12.b	even	2	1		672.2.bk.a		32
21.g	even	6	1		1176.2.c.f		16
21.h	odd	6	1		168.2.bc.a	✓	32
21.h	odd	6	1		1176.2.c.e		16
24.f	even	2	1		672.2.bk.a		32
24.h	odd	2	1		168.2.bc.a	✓	32
28.g	odd	6	1		2016.2.cr.e		32
56.k	odd	6	1		2016.2.cr.e		32
56.p	even	6	1	inner	504.2.cj.e		32
84.j	odd	6	1		4704.2.c.f		16
84.n	even	6	1		672.2.bk.a		32
84.n	even	6	1		4704.2.c.e		16
168.s	odd	6	1		168.2.bc.a	✓	32
168.s	odd	6	1		1176.2.c.e		16
168.v	even	6	1		672.2.bk.a		32
168.v	even	6	1		4704.2.c.e		16
168.ba	even	6	1		1176.2.c.f		16
168.be	odd	6	1		4704.2.c.f		16

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^32 - 48*T5^30 + 1402*T5^28 - 26528*T5^26 + 370859*T5^24 - 3789184*T5^22 + 29482826*T5^20 - 168147456*T5^18 + 724270609*T5^16 - 2228479968*T5^14 + 5037013168*T5^12 - 7430619264*T5^10 + 7421657792*T5^8 - 3023744000*T5^6 + 886791168*T5^4 - 1912832*T5^2 + 4096