Newform orbit 1400.2.x.c

• Label: 1400.2.x.c
• Level: $1400$
• Weight: $2$
• Character orbit: 1400.x
• Analytic conductor: $11.179$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.1790562830\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
657.1		0		−2.09284	−	2.09284i		0			0			0		0.510946	−	2.59595i		0				5.75996i		0
657.2		0		−2.09284	−	2.09284i		0			0			0		2.59595	−	0.510946i		0				5.75996i		0
657.3		0		−1.26512	−	1.26512i		0			0			0		−1.75168	−	1.98283i		0				0.201074i		0
657.4		0		−1.26512	−	1.26512i		0			0			0		1.98283	+	1.75168i		0				0.201074i		0
657.5		0		−0.923076	−	0.923076i		0			0			0		−2.48246	+	0.915096i		0			−	1.29586i		0
657.6		0		−0.923076	−	0.923076i		0			0			0		−0.915096	+	2.48246i		0			−	1.29586i		0
657.7		0		−0.409160	−	0.409160i		0			0			0		0.738062	−	2.54072i		0			−	2.66518i		0
657.8		0		−0.409160	−	0.409160i		0			0			0		2.54072	−	0.738062i		0			−	2.66518i		0
657.9		0		0.409160	+	0.409160i		0			0			0		−2.54072	+	0.738062i		0			−	2.66518i		0
657.10		0		0.409160	+	0.409160i		0			0			0		−0.738062	+	2.54072i		0			−	2.66518i		0
657.11		0		0.923076	+	0.923076i		0			0			0		0.915096	−	2.48246i		0			−	1.29586i		0
657.12		0		0.923076	+	0.923076i		0			0			0		2.48246	−	0.915096i		0			−	1.29586i		0
657.13		0		1.26512	+	1.26512i		0			0			0		−1.98283	−	1.75168i		0				0.201074i		0
657.14		0		1.26512	+	1.26512i		0			0			0		1.75168	+	1.98283i		0				0.201074i		0
657.15		0		2.09284	+	2.09284i		0			0			0		−2.59595	+	0.510946i		0				5.75996i		0
657.16		0		2.09284	+	2.09284i		0			0			0		−0.510946	+	2.59595i		0				5.75996i		0
993.1		0		−2.09284	+	2.09284i		0			0			0		0.510946	+	2.59595i		0			−	5.75996i		0
993.2		0		−2.09284	+	2.09284i		0			0			0		2.59595	+	0.510946i		0			−	5.75996i		0
993.3		0		−1.26512	+	1.26512i		0			0			0		−1.75168	+	1.98283i		0			−	0.201074i		0
993.4		0		−1.26512	+	1.26512i		0			0			0		1.98283	−	1.75168i		0			−	0.201074i		0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1400.2.x.c	✓	32
5.b	even	2	1	inner	1400.2.x.c	✓	32
5.c	odd	4	2	inner	1400.2.x.c	✓	32
7.b	odd	2	1	inner	1400.2.x.c	✓	32
35.c	odd	2	1	inner	1400.2.x.c	✓	32
35.f	even	4	2	inner	1400.2.x.c	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1400.2.x.c	✓	32	1.a	even	1	1	trivial
1400.2.x.c	✓	32	5.b	even	2	1	inner
1400.2.x.c	✓	32	5.c	odd	4	2	inner
1400.2.x.c	✓	32	7.b	odd	2	1	inner
1400.2.x.c	✓	32	35.c	odd	2	1	inner
1400.2.x.c	✓	32	35.f	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 90T_{3}^{12} + 1049T_{3}^{8} + 2400T_{3}^{4} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\).