Newform orbit 735.2.g.c

• Label: 735.2.g.c
• Level: $735$
• Weight: $2$
• Character orbit: 735.g
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} + 40 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
734.1	−2.20906			−1.59632	−	0.672139i	2.87996			1.33217	+	1.79592i	3.52637	+	1.48480i		0		−1.94389			2.09646	+	2.14589i	−2.94284	−	3.96730i
734.2	−2.20906			−1.59632	+	0.672139i	2.87996			1.33217	−	1.79592i	3.52637	−	1.48480i		0		−1.94389			2.09646	−	2.14589i	−2.94284	+	3.96730i
734.3	−2.20906			1.59632	−	0.672139i	2.87996			−1.33217	+	1.79592i	−3.52637	+	1.48480i		0		−1.94389			2.09646	−	2.14589i	2.94284	−	3.96730i
734.4	−2.20906			1.59632	+	0.672139i	2.87996			−1.33217	−	1.79592i	−3.52637	−	1.48480i		0		−1.94389			2.09646	+	2.14589i	2.94284	+	3.96730i
734.5	−2.04548			−0.965325	−	1.43811i	2.18398			2.23143	−	0.144014i	1.97455	+	2.94161i		0		−0.376326			−1.13630	+	2.77648i	−4.56433	+	0.294577i
734.6	−2.04548			−0.965325	+	1.43811i	2.18398			2.23143	+	0.144014i	1.97455	−	2.94161i		0		−0.376326			−1.13630	−	2.77648i	−4.56433	−	0.294577i
734.7	−2.04548			0.965325	−	1.43811i	2.18398			−2.23143	−	0.144014i	−1.97455	+	2.94161i		0		−0.376326			−1.13630	−	2.77648i	4.56433	+	0.294577i
734.8	−2.04548			0.965325	+	1.43811i	2.18398			−2.23143	+	0.144014i	−1.97455	−	2.94161i		0		−0.376326			−1.13630	+	2.77648i	4.56433	−	0.294577i
734.9	−0.903339			−1.72180	−	0.188163i	−1.18398			−1.94641	−	1.10068i	1.55537	+	0.169975i		0		2.87621			2.92919	+	0.647957i	1.75827	+	0.994285i
734.10	−0.903339			−1.72180	+	0.188163i	−1.18398			−1.94641	+	1.10068i	1.55537	−	0.169975i		0		2.87621			2.92919	−	0.647957i	1.75827	−	0.994285i
734.11	−0.903339			1.72180	−	0.188163i	−1.18398			1.94641	−	1.10068i	−1.55537	+	0.169975i		0		2.87621			2.92919	−	0.647957i	−1.75827	+	0.994285i
734.12	−0.903339			1.72180	+	0.188163i	−1.18398			1.94641	+	1.10068i	−1.55537	−	0.169975i		0		2.87621			2.92919	+	0.647957i	−1.75827	−	0.994285i
734.13	−0.346467			−1.43364	−	0.971945i	−1.87996			1.85945	+	1.24195i	0.496709	+	0.336746i		0		1.34428			1.11065	+	2.78684i	−0.644238	−	0.430294i
734.14	−0.346467			−1.43364	+	0.971945i	−1.87996			1.85945	−	1.24195i	0.496709	−	0.336746i		0		1.34428			1.11065	−	2.78684i	−0.644238	+	0.430294i
734.15	−0.346467			1.43364	−	0.971945i	−1.87996			−1.85945	+	1.24195i	−0.496709	+	0.336746i		0		1.34428			1.11065	−	2.78684i	0.644238	−	0.430294i
734.16	−0.346467			1.43364	+	0.971945i	−1.87996			−1.85945	−	1.24195i	−0.496709	−	0.336746i		0		1.34428			1.11065	+	2.78684i	0.644238	+	0.430294i
734.17	0.346467			−1.43364	−	0.971945i	−1.87996			−1.85945	+	1.24195i	−0.496709	−	0.336746i		0		−1.34428			1.11065	+	2.78684i	−0.644238	+	0.430294i
734.18	0.346467			−1.43364	+	0.971945i	−1.87996			−1.85945	−	1.24195i	−0.496709	+	0.336746i		0		−1.34428			1.11065	−	2.78684i	−0.644238	−	0.430294i
734.19	0.346467			1.43364	−	0.971945i	−1.87996			1.85945	+	1.24195i	0.496709	−	0.336746i		0		−1.34428			1.11065	−	2.78684i	0.644238	+	0.430294i
734.20	0.346467			1.43364	+	0.971945i	−1.87996			1.85945	−	1.24195i	0.496709	+	0.336746i		0		−1.34428			1.11065	+	2.78684i	0.644238	−	0.430294i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
15.d	odd	2	1	inner
21.c	even	2	1	inner
35.c	odd	2	1	inner
105.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.g.c	✓	32
3.b	odd	2	1	inner	735.2.g.c	✓	32
5.b	even	2	1	inner	735.2.g.c	✓	32
7.b	odd	2	1	inner	735.2.g.c	✓	32
7.c	even	3	2		735.2.p.g		64
7.d	odd	6	2		735.2.p.g		64
15.d	odd	2	1	inner	735.2.g.c	✓	32
21.c	even	2	1	inner	735.2.g.c	✓	32
21.g	even	6	2		735.2.p.g		64
21.h	odd	6	2		735.2.p.g		64
35.c	odd	2	1	inner	735.2.g.c	✓	32
35.i	odd	6	2		735.2.p.g		64
35.j	even	6	2		735.2.p.g		64
105.g	even	2	1	inner	735.2.g.c	✓	32
105.o	odd	6	2		735.2.p.g		64
105.p	even	6	2		735.2.p.g		64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.g.c	✓	32	1.a	even	1	1	trivial
735.2.g.c	✓	32	3.b	odd	2	1	inner
735.2.g.c	✓	32	5.b	even	2	1	inner
735.2.g.c	✓	32	7.b	odd	2	1	inner
735.2.g.c	✓	32	15.d	odd	2	1	inner
735.2.g.c	✓	32	21.c	even	2	1	inner
735.2.g.c	✓	32	35.c	odd	2	1	inner
735.2.g.c	✓	32	105.g	even	2	1	inner
735.2.p.g		64	7.c	even	3	2
735.2.p.g		64	7.d	odd	6	2
735.2.p.g		64	21.g	even	6	2
735.2.p.g		64	21.h	odd	6	2
735.2.p.g		64	35.i	odd	6	2
735.2.p.g		64	35.j	even	6	2
735.2.p.g		64	105.o	odd	6	2
735.2.p.g		64	105.p	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 10T_{2}^{6} + 29T_{2}^{4} - 20T_{2}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\).