Newform orbit 1600.2.s.e

• Label: 1600.2.s.e
• Level: $1600$
• Weight: $2$
• Character orbit: 1600.s
• Analytic conductor: $12.776$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.s (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.7760643234\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 400)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 40 q^{9} + 20 q^{11} - 12 q^{19} - 8 q^{29} - 20 q^{51} + 8 q^{59} - 48 q^{61} + 64 q^{69} + 16 q^{71} - 104 q^{79} + 48 q^{81} - 96 q^{89} - 64 q^{91} + 128 q^{99}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
207.1		0		−3.25766				0			0			0		2.54012	+	2.54012i		0		7.61238				0
207.2		0		−2.70780				0			0			0		−1.60911	−	1.60911i		0		4.33218				0
207.3		0		−2.35800				0			0			0		−2.66357	−	2.66357i		0		2.56018				0
207.4		0		−1.86755				0			0			0		0.719989	+	0.719989i		0		0.487737				0
207.5		0		−0.790153				0			0			0		−0.139907	−	0.139907i		0		−2.37566				0
207.6		0		−0.619018				0			0			0		1.82373	+	1.82373i		0		−2.61682				0
207.7		0		0.619018				0			0			0		−1.82373	−	1.82373i		0		−2.61682				0
207.8		0		0.790153				0			0			0		0.139907	+	0.139907i		0		−2.37566				0
207.9		0		1.86755				0			0			0		−0.719989	−	0.719989i		0		0.487737				0
207.10		0		2.35800				0			0			0		2.66357	+	2.66357i		0		2.56018				0
207.11		0		2.70780				0			0			0		1.60911	+	1.60911i		0		4.33218				0
207.12		0		3.25766				0			0			0		−2.54012	−	2.54012i		0		7.61238				0
943.1		0		−3.25766				0			0			0		2.54012	−	2.54012i		0		7.61238				0
943.2		0		−2.70780				0			0			0		−1.60911	+	1.60911i		0		4.33218				0
943.3		0		−2.35800				0			0			0		−2.66357	+	2.66357i		0		2.56018				0
943.4		0		−1.86755				0			0			0		0.719989	−	0.719989i		0		0.487737				0
943.5		0		−0.790153				0			0			0		−0.139907	+	0.139907i		0		−2.37566				0
943.6		0		−0.619018				0			0			0		1.82373	−	1.82373i		0		−2.61682				0
943.7		0		0.619018				0			0			0		−1.82373	+	1.82373i		0		−2.61682				0
943.8		0		0.790153				0			0			0		0.139907	−	0.139907i		0		−2.37566				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
80.j	even	4	1	inner
80.s	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1600.2.s.e		24
4.b	odd	2	1		400.2.s.e	yes	24
5.b	even	2	1	inner	1600.2.s.e		24
5.c	odd	4	2		1600.2.j.e		24
16.e	even	4	1		400.2.j.e	✓	24
16.f	odd	4	1		1600.2.j.e		24
20.d	odd	2	1		400.2.s.e	yes	24
20.e	even	4	2		400.2.j.e	✓	24
80.i	odd	4	1		400.2.s.e	yes	24
80.j	even	4	1	inner	1600.2.s.e		24
80.k	odd	4	1		1600.2.j.e		24
80.q	even	4	1		400.2.j.e	✓	24
80.s	even	4	1	inner	1600.2.s.e		24
80.t	odd	4	1		400.2.s.e	yes	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
400.2.j.e	✓	24	16.e	even	4	1
400.2.j.e	✓	24	20.e	even	4	2
400.2.j.e	✓	24	80.q	even	4	1
400.2.s.e	yes	24	4.b	odd	2	1
400.2.s.e	yes	24	20.d	odd	2	1
400.2.s.e	yes	24	80.i	odd	4	1
400.2.s.e	yes	24	80.t	odd	4	1
1600.2.j.e		24	5.c	odd	4	2
1600.2.j.e		24	16.f	odd	4	1
1600.2.j.e		24	80.k	odd	4	1
1600.2.s.e		24	1.a	even	1	1	trivial
1600.2.s.e		24	5.b	even	2	1	inner
1600.2.s.e		24	80.j	even	4	1	inner
1600.2.s.e		24	80.s	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 28T_{3}^{10} + 287T_{3}^{8} - 1320T_{3}^{6} + 2631T_{3}^{4} - 1772T_{3}^{2} + 361 \) acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\).