Newform orbit 400.6.n.h

• Label: 400.6.n.h
• Level: $400$
• Weight: $6$
• Character orbit: 400.n
• Analytic conductor: $64.154$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(64.1535279252\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 19936 q^{21} + 32568 q^{41} + 454944 q^{61} + 1059176 q^{81}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
143.1		0		−13.8504	+	13.8504i		0			0			0		−156.283	−	156.283i		0			−	140.668i		0
143.2		0		−13.8504	+	13.8504i		0			0			0		−156.283	−	156.283i		0			−	140.668i		0
143.3		0		−13.2732	+	13.2732i		0			0			0		25.6490	+	25.6490i		0			−	109.358i		0
143.4		0		−13.2732	+	13.2732i		0			0			0		25.6490	+	25.6490i		0			−	109.358i		0
143.5		0		−5.78678	+	5.78678i		0			0			0		99.9080	+	99.9080i		0				176.026i		0
143.6		0		−5.78678	+	5.78678i		0			0			0		99.9080	+	99.9080i		0				176.026i		0
143.7		0		5.78678	−	5.78678i		0			0			0		−99.9080	−	99.9080i		0				176.026i		0
143.8		0		5.78678	−	5.78678i		0			0			0		−99.9080	−	99.9080i		0				176.026i		0
143.9		0		13.2732	−	13.2732i		0			0			0		−25.6490	−	25.6490i		0			−	109.358i		0
143.10		0		13.2732	−	13.2732i		0			0			0		−25.6490	−	25.6490i		0			−	109.358i		0
143.11		0		13.8504	−	13.8504i		0			0			0		156.283	+	156.283i		0			−	140.668i		0
143.12		0		13.8504	−	13.8504i		0			0			0		156.283	+	156.283i		0			−	140.668i		0
207.1		0		−13.8504	−	13.8504i		0			0			0		−156.283	+	156.283i		0				140.668i		0
207.2		0		−13.8504	−	13.8504i		0			0			0		−156.283	+	156.283i		0				140.668i		0
207.3		0		−13.2732	−	13.2732i		0			0			0		25.6490	−	25.6490i		0				109.358i		0
207.4		0		−13.2732	−	13.2732i		0			0			0		25.6490	−	25.6490i		0				109.358i		0
207.5		0		−5.78678	−	5.78678i		0			0			0		99.9080	−	99.9080i		0			−	176.026i		0
207.6		0		−5.78678	−	5.78678i		0			0			0		99.9080	−	99.9080i		0			−	176.026i		0
207.7		0		5.78678	+	5.78678i		0			0			0		−99.9080	+	99.9080i		0			−	176.026i		0
207.8		0		5.78678	+	5.78678i		0			0			0		−99.9080	+	99.9080i		0			−	176.026i		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
5.c	odd	4	2	inner
20.d	odd	2	1	inner
20.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.6.n.h	✓	24
4.b	odd	2	1	inner	400.6.n.h	✓	24
5.b	even	2	1	inner	400.6.n.h	✓	24
5.c	odd	4	2	inner	400.6.n.h	✓	24
20.d	odd	2	1	inner	400.6.n.h	✓	24
20.e	even	4	2	inner	400.6.n.h	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
400.6.n.h	✓	24	1.a	even	1	1	trivial
400.6.n.h	✓	24	4.b	odd	2	1	inner
400.6.n.h	✓	24	5.b	even	2	1	inner
400.6.n.h	✓	24	5.c	odd	4	2	inner
400.6.n.h	✓	24	20.d	odd	2	1	inner
400.6.n.h	✓	24	20.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 275843T_{3}^{8} + 19493123203T_{3}^{4} + 81976382754561 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).