Newform orbit 735.2.m.c

• Label: 735.2.m.c
• Level: $735$
• Weight: $2$
• Character orbit: 735.m
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
97.1	−1.59147	−	1.59147i	−0.707107	−	0.707107i			3.06556i	1.90623	+	1.16889i			2.25068i		0		1.69581	−	1.69581i			1.00000i	−1.17345	−	4.89396i
97.2	−1.59147	−	1.59147i	0.707107	+	0.707107i			3.06556i	−1.90623	−	1.16889i		−	2.25068i		0		1.69581	−	1.69581i			1.00000i	1.17345	+	4.89396i
97.3	−1.25995	−	1.25995i	−0.707107	−	0.707107i			1.17495i	−0.0279443	−	2.23589i			1.78184i		0		−1.03952	+	1.03952i			1.00000i	−2.78191	+	2.85232i
97.4	−1.25995	−	1.25995i	0.707107	+	0.707107i			1.17495i	0.0279443	+	2.23589i		−	1.78184i		0		−1.03952	+	1.03952i			1.00000i	2.78191	−	2.85232i
97.5	−1.09556	−	1.09556i	−0.707107	−	0.707107i			0.400511i	−0.531984	+	2.17186i			1.54936i		0		−1.75234	+	1.75234i			1.00000i	2.96223	−	1.79659i
97.6	−1.09556	−	1.09556i	0.707107	+	0.707107i			0.400511i	0.531984	−	2.17186i		−	1.54936i		0		−1.75234	+	1.75234i			1.00000i	−2.96223	+	1.79659i
97.7	−0.474562	−	0.474562i	−0.707107	−	0.707107i		−	1.54958i	−0.484013	−	2.18306i			0.671132i		0		−1.68450	+	1.68450i			1.00000i	−0.806302	+	1.26569i
97.8	−0.474562	−	0.474562i	0.707107	+	0.707107i		−	1.54958i	0.484013	+	2.18306i		−	0.671132i		0		−1.68450	+	1.68450i			1.00000i	0.806302	−	1.26569i
97.9	0.288785	+	0.288785i	−0.707107	−	0.707107i		−	1.83321i	1.48985	+	1.66743i		−	0.408404i		0		1.10697	−	1.10697i			1.00000i	−0.0512822	+	0.911777i
97.10	0.288785	+	0.288785i	0.707107	+	0.707107i		−	1.83321i	−1.48985	−	1.66743i			0.408404i		0		1.10697	−	1.10697i			1.00000i	0.0512822	−	0.911777i
97.11	0.709756	+	0.709756i	−0.707107	−	0.707107i		−	0.992492i	2.20896	−	0.347130i		−	1.00375i		0		2.12394	−	2.12394i			1.00000i	1.81420	+	1.32145i
97.12	0.709756	+	0.709756i	0.707107	+	0.707107i		−	0.992492i	−2.20896	+	0.347130i			1.00375i		0		2.12394	−	2.12394i			1.00000i	−1.81420	−	1.32145i
97.13	1.64576	+	1.64576i	−0.707107	−	0.707107i			3.41703i	−0.941924	−	2.02800i		−	2.32745i		0		−2.33208	+	2.33208i			1.00000i	1.78741	−	4.88777i
97.14	1.64576	+	1.64576i	0.707107	+	0.707107i			3.41703i	0.941924	+	2.02800i			2.32745i		0		−2.33208	+	2.33208i			1.00000i	−1.78741	+	4.88777i
97.15	1.77725	+	1.77725i	−0.707107	−	0.707107i			4.31723i	−2.20496	+	0.371678i		−	2.51341i		0		−4.11829	+	4.11829i			1.00000i	−4.57933	−	3.25820i
97.16	1.77725	+	1.77725i	0.707107	+	0.707107i			4.31723i	2.20496	−	0.371678i			2.51341i		0		−4.11829	+	4.11829i			1.00000i	4.57933	+	3.25820i
538.1	−1.59147	+	1.59147i	−0.707107	+	0.707107i		−	3.06556i	1.90623	−	1.16889i		−	2.25068i		0		1.69581	+	1.69581i		−	1.00000i	−1.17345	+	4.89396i
538.2	−1.59147	+	1.59147i	0.707107	−	0.707107i		−	3.06556i	−1.90623	+	1.16889i			2.25068i		0		1.69581	+	1.69581i		−	1.00000i	1.17345	−	4.89396i
538.3	−1.25995	+	1.25995i	−0.707107	+	0.707107i		−	1.17495i	−0.0279443	+	2.23589i		−	1.78184i		0		−1.03952	−	1.03952i		−	1.00000i	−2.78191	−	2.85232i
538.4	−1.25995	+	1.25995i	0.707107	−	0.707107i		−	1.17495i	0.0279443	−	2.23589i			1.78184i		0		−1.03952	−	1.03952i		−	1.00000i	2.78191	+	2.85232i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
7.b	odd	2	1	inner
35.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.m.c		32
5.c	odd	4	1	inner	735.2.m.c		32
7.b	odd	2	1	inner	735.2.m.c		32
7.c	even	3	1		105.2.u.a	✓	32
7.c	even	3	1		735.2.v.b		32
7.d	odd	6	1		105.2.u.a	✓	32
7.d	odd	6	1		735.2.v.b		32
21.g	even	6	1		315.2.bz.d		32
21.h	odd	6	1		315.2.bz.d		32
35.f	even	4	1	inner	735.2.m.c		32
35.i	odd	6	1		525.2.bc.e		32
35.j	even	6	1		525.2.bc.e		32
35.k	even	12	1		105.2.u.a	✓	32
35.k	even	12	1		525.2.bc.e		32
35.k	even	12	1		735.2.v.b		32
35.l	odd	12	1		105.2.u.a	✓	32
35.l	odd	12	1		525.2.bc.e		32
35.l	odd	12	1		735.2.v.b		32
105.w	odd	12	1		315.2.bz.d		32
105.x	even	12	1		315.2.bz.d		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.u.a	✓	32	7.c	even	3	1
105.2.u.a	✓	32	7.d	odd	6	1
105.2.u.a	✓	32	35.k	even	12	1
105.2.u.a	✓	32	35.l	odd	12	1
315.2.bz.d		32	21.g	even	6	1
315.2.bz.d		32	21.h	odd	6	1
315.2.bz.d		32	105.w	odd	12	1
315.2.bz.d		32	105.x	even	12	1
525.2.bc.e		32	35.i	odd	6	1
525.2.bc.e		32	35.j	even	6	1
525.2.bc.e		32	35.k	even	12	1
525.2.bc.e		32	35.l	odd	12	1
735.2.m.c		32	1.a	even	1	1	trivial
735.2.m.c		32	5.c	odd	4	1	inner
735.2.m.c		32	7.b	odd	2	1	inner
735.2.m.c		32	35.f	even	4	1	inner
735.2.v.b		32	7.c	even	3	1
735.2.v.b		32	7.d	odd	6	1
735.2.v.b		32	35.k	even	12	1
735.2.v.b		32	35.l	odd	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

T2^16 + 4*T2^13 + 56*T2^12 + 24*T2^11 + 8*T2^10 + 96*T2^9 + 852*T2^8 + 680*T2^7 + 224*T2^6 - 360*T2^5 + 1236*T2^4 + 496*T2^3 + 128*T2^2 - 160*T2 + 100

T13^32 + 3752*T13^28 + 4933436*T13^24 + 2838285760*T13^20 + 768155582310*T13^16 + 95927004521096*T13^12 + 4450500111334252*T13^8 + 2012513580680016*T13^4 + 9926437890625