Newform orbit 1176.2.p.a

• Label: 1176.2.p.a
• Level: $1176$
• Weight: $2$
• Character orbit: 1176.p
• Analytic conductor: $9.390$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.39040727770\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
979.1	−1.32020	−	0.507016i			1.00000i	1.48587	+	1.33873i	0.451079			0.507016	−	1.32020i		0		−1.28289	−	2.52075i	−1.00000			−0.595516	−	0.228704i
979.2	−1.32020	−	0.507016i		−	1.00000i	1.48587	+	1.33873i	−0.451079			−0.507016	+	1.32020i		0		−1.28289	−	2.52075i	−1.00000			0.595516	+	0.228704i
979.3	−1.32020	+	0.507016i		−	1.00000i	1.48587	−	1.33873i	0.451079			0.507016	+	1.32020i		0		−1.28289	+	2.52075i	−1.00000			−0.595516	+	0.228704i
979.4	−1.32020	+	0.507016i			1.00000i	1.48587	−	1.33873i	−0.451079			−0.507016	−	1.32020i		0		−1.28289	+	2.52075i	−1.00000			0.595516	−	0.228704i
979.5	−1.04886	−	0.948630i			1.00000i	0.200203	+	1.98995i	2.88284			0.948630	−	1.04886i		0		1.67775	−	2.27710i	−1.00000			−3.02369	−	2.73475i
979.6	−1.04886	−	0.948630i		−	1.00000i	0.200203	+	1.98995i	−2.88284			−0.948630	+	1.04886i		0		1.67775	−	2.27710i	−1.00000			3.02369	+	2.73475i
979.7	−1.04886	+	0.948630i		−	1.00000i	0.200203	−	1.98995i	2.88284			0.948630	+	1.04886i		0		1.67775	+	2.27710i	−1.00000			−3.02369	+	2.73475i
979.8	−1.04886	+	0.948630i			1.00000i	0.200203	−	1.98995i	−2.88284			−0.948630	−	1.04886i		0		1.67775	+	2.27710i	−1.00000			3.02369	−	2.73475i
979.9	−0.688575	−	1.23526i		−	1.00000i	−1.05173	+	1.70114i	2.28105			−1.23526	+	0.688575i		0		2.82554	+	0.127796i	−1.00000			−1.57068	−	2.81769i
979.10	−0.688575	−	1.23526i			1.00000i	−1.05173	+	1.70114i	−2.28105			1.23526	−	0.688575i		0		2.82554	+	0.127796i	−1.00000			1.57068	+	2.81769i
979.11	−0.688575	+	1.23526i			1.00000i	−1.05173	−	1.70114i	2.28105			−1.23526	−	0.688575i		0		2.82554	−	0.127796i	−1.00000			−1.57068	+	2.81769i
979.12	−0.688575	+	1.23526i		−	1.00000i	−1.05173	−	1.70114i	−2.28105			1.23526	+	0.688575i		0		2.82554	−	0.127796i	−1.00000			1.57068	−	2.81769i
979.13	−0.298688	−	1.38231i			1.00000i	−1.82157	+	0.825759i	0.310560			1.38231	−	0.298688i		0		1.68554	+	2.27133i	−1.00000			−0.0927604	−	0.429290i
979.14	−0.298688	−	1.38231i		−	1.00000i	−1.82157	+	0.825759i	−0.310560			−1.38231	+	0.298688i		0		1.68554	+	2.27133i	−1.00000			0.0927604	+	0.429290i
979.15	−0.298688	+	1.38231i		−	1.00000i	−1.82157	−	0.825759i	0.310560			1.38231	+	0.298688i		0		1.68554	−	2.27133i	−1.00000			−0.0927604	+	0.429290i
979.16	−0.298688	+	1.38231i			1.00000i	−1.82157	−	0.825759i	−0.310560			−1.38231	−	0.298688i		0		1.68554	−	2.27133i	−1.00000			0.0927604	−	0.429290i
979.17	0.765334	−	1.18923i			1.00000i	−0.828528	−	1.82031i	4.17553			1.18923	+	0.765334i		0		−2.79887	−	0.407838i	−1.00000			3.19567	−	4.96566i
979.18	0.765334	−	1.18923i		−	1.00000i	−0.828528	−	1.82031i	−4.17553			−1.18923	−	0.765334i		0		−2.79887	−	0.407838i	−1.00000			−3.19567	+	4.96566i
979.19	0.765334	+	1.18923i		−	1.00000i	−0.828528	+	1.82031i	4.17553			1.18923	−	0.765334i		0		−2.79887	+	0.407838i	−1.00000			3.19567	+	4.96566i
979.20	0.765334	+	1.18923i			1.00000i	−0.828528	+	1.82031i	−4.17553			−1.18923	+	0.765334i		0		−2.79887	+	0.407838i	−1.00000			−3.19567	−	4.96566i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
8.d	odd	2	1	inner
56.e	even	2	1	inner

Twists

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

    

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