Newform orbit 1680.2.q.c

• Label: 1680.2.q.c
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.q
• Analytic conductor: $13.415$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.q (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4148675396\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{9} + 8 q^{21} - 32 q^{25} + 16 q^{49} + 48 q^{65} + 32 q^{81} - 64 q^{85}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
559.1		0			−	1.00000i		0		−1.58345	−	1.57882i		0		2.37980	−	1.15610i		0		−1.00000				0
559.2		0			−	1.00000i		0		1.58345	−	1.57882i		0		2.37980	−	1.15610i		0		−1.00000				0
559.3		0				1.00000i		0		−1.58345	+	1.57882i		0		2.37980	+	1.15610i		0		−1.00000				0
559.4		0				1.00000i		0		1.58345	+	1.57882i		0		2.37980	+	1.15610i		0		−1.00000				0
559.5		0			−	1.00000i		0		−1.58345	+	1.57882i		0		−2.37980	−	1.15610i		0		−1.00000				0
559.6		0			−	1.00000i		0		1.58345	+	1.57882i		0		−2.37980	−	1.15610i		0		−1.00000				0
559.7		0				1.00000i		0		−1.58345	−	1.57882i		0		−2.37980	+	1.15610i		0		−1.00000				0
559.8		0				1.00000i		0		1.58345	−	1.57882i		0		−2.37980	+	1.15610i		0		−1.00000				0
559.9		0			−	1.00000i		0		−0.890213	+	2.05122i		0		0.425671	+	2.61128i		0		−1.00000				0
559.10		0			−	1.00000i		0		0.890213	+	2.05122i		0		0.425671	+	2.61128i		0		−1.00000				0
559.11		0				1.00000i		0		−0.890213	−	2.05122i		0		0.425671	−	2.61128i		0		−1.00000				0
559.12		0				1.00000i		0		0.890213	−	2.05122i		0		0.425671	−	2.61128i		0		−1.00000				0
559.13		0			−	1.00000i		0		−2.14256	−	0.639872i		0		2.29750	+	1.31206i		0		−1.00000				0
559.14		0			−	1.00000i		0		2.14256	−	0.639872i		0		2.29750	+	1.31206i		0		−1.00000				0
559.15		0				1.00000i		0		−2.14256	+	0.639872i		0		2.29750	−	1.31206i		0		−1.00000				0
559.16		0				1.00000i		0		2.14256	+	0.639872i		0		2.29750	−	1.31206i		0		−1.00000				0
559.17		0			−	1.00000i		0		−0.890213	−	2.05122i		0		−0.425671	+	2.61128i		0		−1.00000				0
559.18		0			−	1.00000i		0		0.890213	−	2.05122i		0		−0.425671	+	2.61128i		0		−1.00000				0
559.19		0				1.00000i		0		−0.890213	+	2.05122i		0		−0.425671	−	2.61128i		0		−1.00000				0
559.20		0				1.00000i		0		0.890213	+	2.05122i		0		−0.425671	−	2.61128i		0		−1.00000				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
20.d	odd	2	1	inner
28.d	even	2	1	inner
35.c	odd	2	1	inner
140.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1680.2.q.c	✓	32
4.b	odd	2	1	inner	1680.2.q.c	✓	32
5.b	even	2	1	inner	1680.2.q.c	✓	32
7.b	odd	2	1	inner	1680.2.q.c	✓	32
20.d	odd	2	1	inner	1680.2.q.c	✓	32
28.d	even	2	1	inner	1680.2.q.c	✓	32
35.c	odd	2	1	inner	1680.2.q.c	✓	32
140.c	even	2	1	inner	1680.2.q.c	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1680.2.q.c	✓	32	1.a	even	1	1	trivial
1680.2.q.c	✓	32	4.b	odd	2	1	inner
1680.2.q.c	✓	32	5.b	even	2	1	inner
1680.2.q.c	✓	32	7.b	odd	2	1	inner
1680.2.q.c	✓	32	20.d	odd	2	1	inner
1680.2.q.c	✓	32	28.d	even	2	1	inner
1680.2.q.c	✓	32	35.c	odd	2	1	inner
1680.2.q.c	✓	32	140.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 44T_{11}^{6} + 288T_{11}^{4} + 512T_{11}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\).