Newform orbit 735.2.g.b

• Label: 735.2.g.b
• Level: $735$
• Weight: $2$
• Character orbit: 735.g
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $24$
• 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:	\(24\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} + 12 q^{9} - 24 q^{15} + 24 q^{16} + 24 q^{25} - 36 q^{30} + 84 q^{36} + 24 q^{39} - 72 q^{46} + 24 q^{51} - 24 q^{60} + 24 q^{64} - 96 q^{79} + 12 q^{81} + 48 q^{85} - 48 q^{99}+O(q^{100}) \)

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	−2.50798			−1.47954	−	0.900534i	4.28995			−1.94085	−	1.11045i	3.71065	+	2.25852i		0		−5.74313			1.37808	+	2.66475i	4.86760	+	2.78499i
734.2	−2.50798			−1.47954	+	0.900534i	4.28995			−1.94085	+	1.11045i	3.71065	−	2.25852i		0		−5.74313			1.37808	−	2.66475i	4.86760	−	2.78499i
734.3	−2.50798			1.47954	−	0.900534i	4.28995			1.94085	−	1.11045i	−3.71065	+	2.25852i		0		−5.74313			1.37808	−	2.66475i	−4.86760	+	2.78499i
734.4	−2.50798			1.47954	+	0.900534i	4.28995			1.94085	+	1.11045i	−3.71065	−	2.25852i		0		−5.74313			1.37808	+	2.66475i	−4.86760	−	2.78499i
734.5	−1.51469			−1.66512	−	0.476833i	0.294280			0.775809	−	2.09717i	2.52214	+	0.722254i		0		2.58363			2.54526	+	1.58797i	−1.17511	+	3.17656i
734.6	−1.51469			−1.66512	+	0.476833i	0.294280			0.775809	+	2.09717i	2.52214	−	0.722254i		0		2.58363			2.54526	−	1.58797i	−1.17511	−	3.17656i
734.7	−1.51469			1.66512	−	0.476833i	0.294280			−0.775809	−	2.09717i	−2.52214	+	0.722254i		0		2.58363			2.54526	−	1.58797i	1.17511	+	3.17656i
734.8	−1.51469			1.66512	+	0.476833i	0.294280			−0.775809	+	2.09717i	−2.52214	−	0.722254i		0		2.58363			2.54526	+	1.58797i	1.17511	−	3.17656i
734.9	−0.644806			−0.536966	−	1.64671i	−1.58423			2.15203	−	0.607270i	0.346239	+	1.06181i		0		2.31113			−2.42334	+	1.76846i	−1.38764	+	0.391571i
734.10	−0.644806			−0.536966	+	1.64671i	−1.58423			2.15203	+	0.607270i	0.346239	−	1.06181i		0		2.31113			−2.42334	−	1.76846i	−1.38764	−	0.391571i
734.11	−0.644806			0.536966	−	1.64671i	−1.58423			−2.15203	−	0.607270i	−0.346239	+	1.06181i		0		2.31113			−2.42334	−	1.76846i	1.38764	+	0.391571i
734.12	−0.644806			0.536966	+	1.64671i	−1.58423			−2.15203	+	0.607270i	−0.346239	−	1.06181i		0		2.31113			−2.42334	+	1.76846i	1.38764	−	0.391571i
734.13	0.644806			−0.536966	−	1.64671i	−1.58423			−2.15203	−	0.607270i	−0.346239	−	1.06181i		0		−2.31113			−2.42334	+	1.76846i	−1.38764	−	0.391571i
734.14	0.644806			−0.536966	+	1.64671i	−1.58423			−2.15203	+	0.607270i	−0.346239	+	1.06181i		0		−2.31113			−2.42334	−	1.76846i	−1.38764	+	0.391571i
734.15	0.644806			0.536966	−	1.64671i	−1.58423			2.15203	−	0.607270i	0.346239	−	1.06181i		0		−2.31113			−2.42334	−	1.76846i	1.38764	−	0.391571i
734.16	0.644806			0.536966	+	1.64671i	−1.58423			2.15203	+	0.607270i	0.346239	+	1.06181i		0		−2.31113			−2.42334	+	1.76846i	1.38764	+	0.391571i
734.17	1.51469			−1.66512	−	0.476833i	0.294280			−0.775809	−	2.09717i	−2.52214	−	0.722254i		0		−2.58363			2.54526	+	1.58797i	−1.17511	−	3.17656i
734.18	1.51469			−1.66512	+	0.476833i	0.294280			−0.775809	+	2.09717i	−2.52214	+	0.722254i		0		−2.58363			2.54526	−	1.58797i	−1.17511	+	3.17656i
734.19	1.51469			1.66512	−	0.476833i	0.294280			0.775809	−	2.09717i	2.52214	−	0.722254i		0		−2.58363			2.54526	−	1.58797i	1.17511	−	3.17656i
734.20	1.51469			1.66512	+	0.476833i	0.294280			0.775809	+	2.09717i	2.52214	+	0.722254i		0		−2.58363			2.54526	+	1.58797i	1.17511	+	3.17656i

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.b		24
3.b	odd	2	1	inner	735.2.g.b		24
5.b	even	2	1	inner	735.2.g.b		24
7.b	odd	2	1	inner	735.2.g.b		24
7.c	even	3	1		105.2.p.a	✓	24
7.c	even	3	1		735.2.p.f		24
7.d	odd	6	1		105.2.p.a	✓	24
7.d	odd	6	1		735.2.p.f		24
15.d	odd	2	1	inner	735.2.g.b		24
21.c	even	2	1	inner	735.2.g.b		24
21.g	even	6	1		105.2.p.a	✓	24
21.g	even	6	1		735.2.p.f		24
21.h	odd	6	1		105.2.p.a	✓	24
21.h	odd	6	1		735.2.p.f		24
35.c	odd	2	1	inner	735.2.g.b		24
35.i	odd	6	1		105.2.p.a	✓	24
35.i	odd	6	1		735.2.p.f		24
35.j	even	6	1		105.2.p.a	✓	24
35.j	even	6	1		735.2.p.f		24
35.k	even	12	2		525.2.t.j		24
35.l	odd	12	2		525.2.t.j		24
105.g	even	2	1	inner	735.2.g.b		24
105.o	odd	6	1		105.2.p.a	✓	24
105.o	odd	6	1		735.2.p.f		24
105.p	even	6	1		105.2.p.a	✓	24
105.p	even	6	1		735.2.p.f		24
105.w	odd	12	2		525.2.t.j		24
105.x	even	12	2		525.2.t.j		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.p.a	✓	24	7.c	even	3	1
105.2.p.a	✓	24	7.d	odd	6	1
105.2.p.a	✓	24	21.g	even	6	1
105.2.p.a	✓	24	21.h	odd	6	1
105.2.p.a	✓	24	35.i	odd	6	1
105.2.p.a	✓	24	35.j	even	6	1
105.2.p.a	✓	24	105.o	odd	6	1
105.2.p.a	✓	24	105.p	even	6	1
525.2.t.j		24	35.k	even	12	2
525.2.t.j		24	35.l	odd	12	2
525.2.t.j		24	105.w	odd	12	2
525.2.t.j		24	105.x	even	12	2
735.2.g.b		24	1.a	even	1	1	trivial
735.2.g.b		24	3.b	odd	2	1	inner
735.2.g.b		24	5.b	even	2	1	inner
735.2.g.b		24	7.b	odd	2	1	inner
735.2.g.b		24	15.d	odd	2	1	inner
735.2.g.b		24	21.c	even	2	1	inner
735.2.g.b		24	35.c	odd	2	1	inner
735.2.g.b		24	105.g	even	2	1	inner
735.2.p.f		24	7.c	even	3	1
735.2.p.f		24	7.d	odd	6	1
735.2.p.f		24	21.g	even	6	1
735.2.p.f		24	21.h	odd	6	1
735.2.p.f		24	35.i	odd	6	1
735.2.p.f		24	35.j	even	6	1
735.2.p.f		24	105.o	odd	6	1
735.2.p.f		24	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 9T_{2}^{4} + 18T_{2}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\).