Newform orbit 400.2.s.e

• Label: 400.2.s.e
• Level: $400$
• Weight: $2$
• Character orbit: 400.s
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.19401608085\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
107.1	−1.41248	−	0.0699536i	−0.790153			1.99021	+	0.197616i		0		1.11608	+	0.0552740i	−0.139907	−	0.139907i	−2.79732	−	0.418352i	−2.37566				0
107.2	−1.36763	+	0.359994i	−1.86755			1.74081	−	0.984676i		0		2.55411	−	0.672307i	0.719989	+	0.719989i	−2.02630	+	1.97335i	0.487737				0
107.3	−1.16305	+	0.804554i	2.70780			0.705387	−	1.87148i		0		−3.14932	+	2.17857i	1.60911	+	1.60911i	0.685302	+	2.74415i	4.33218				0
107.4	−1.08097	−	0.911865i	0.619018			0.337006	+	1.97140i		0		−0.669142	−	0.564461i	−1.82373	−	1.82373i	1.43336	−	2.43834i	−2.61682				0
107.5	−0.622050	+	1.27006i	−3.25766			−1.22611	−	1.58008i		0		2.02643	−	4.13743i	2.54012	+	2.54012i	2.76950	−	0.574339i	7.61238				0
107.6	−0.475759	−	1.33179i	−2.35800			−1.54731	+	1.26722i		0		1.12184	+	3.14035i	−2.66357	−	2.66357i	2.42381	+	1.45779i	2.56018				0
107.7	0.475759	+	1.33179i	2.35800			−1.54731	+	1.26722i		0		1.12184	+	3.14035i	2.66357	+	2.66357i	−2.42381	−	1.45779i	2.56018				0
107.8	0.622050	−	1.27006i	3.25766			−1.22611	−	1.58008i		0		2.02643	−	4.13743i	−2.54012	−	2.54012i	−2.76950	+	0.574339i	7.61238				0
107.9	1.08097	+	0.911865i	−0.619018			0.337006	+	1.97140i		0		−0.669142	−	0.564461i	1.82373	+	1.82373i	−1.43336	+	2.43834i	−2.61682				0
107.10	1.16305	−	0.804554i	−2.70780			0.705387	−	1.87148i		0		−3.14932	+	2.17857i	−1.60911	−	1.60911i	−0.685302	−	2.74415i	4.33218				0
107.11	1.36763	−	0.359994i	1.86755			1.74081	−	0.984676i		0		2.55411	−	0.672307i	−0.719989	−	0.719989i	2.02630	−	1.97335i	0.487737				0
107.12	1.41248	+	0.0699536i	0.790153			1.99021	+	0.197616i		0		1.11608	+	0.0552740i	0.139907	+	0.139907i	2.79732	+	0.418352i	−2.37566				0
243.1	−1.41248	+	0.0699536i	−0.790153			1.99021	−	0.197616i		0		1.11608	−	0.0552740i	−0.139907	+	0.139907i	−2.79732	+	0.418352i	−2.37566				0
243.2	−1.36763	−	0.359994i	−1.86755			1.74081	+	0.984676i		0		2.55411	+	0.672307i	0.719989	−	0.719989i	−2.02630	−	1.97335i	0.487737				0
243.3	−1.16305	−	0.804554i	2.70780			0.705387	+	1.87148i		0		−3.14932	−	2.17857i	1.60911	−	1.60911i	0.685302	−	2.74415i	4.33218				0
243.4	−1.08097	+	0.911865i	0.619018			0.337006	−	1.97140i		0		−0.669142	+	0.564461i	−1.82373	+	1.82373i	1.43336	+	2.43834i	−2.61682				0
243.5	−0.622050	−	1.27006i	−3.25766			−1.22611	+	1.58008i		0		2.02643	+	4.13743i	2.54012	−	2.54012i	2.76950	+	0.574339i	7.61238				0
243.6	−0.475759	+	1.33179i	−2.35800			−1.54731	−	1.26722i		0		1.12184	−	3.14035i	−2.66357	+	2.66357i	2.42381	−	1.45779i	2.56018				0
243.7	0.475759	−	1.33179i	2.35800			−1.54731	−	1.26722i		0		1.12184	−	3.14035i	2.66357	−	2.66357i	−2.42381	+	1.45779i	2.56018				0
243.8	0.622050	+	1.27006i	3.25766			−1.22611	+	1.58008i		0		2.02643	+	4.13743i	−2.54012	+	2.54012i	−2.76950	−	0.574339i	7.61238				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	400.2.s.e	yes	24
4.b	odd	2	1		1600.2.s.e		24
5.b	even	2	1	inner	400.2.s.e	yes	24
5.c	odd	4	2		400.2.j.e	✓	24
16.e	even	4	1		1600.2.j.e		24
16.f	odd	4	1		400.2.j.e	✓	24
20.d	odd	2	1		1600.2.s.e		24
20.e	even	4	2		1600.2.j.e		24
80.i	odd	4	1		1600.2.s.e		24
80.j	even	4	1	inner	400.2.s.e	yes	24
80.k	odd	4	1		400.2.j.e	✓	24
80.q	even	4	1		1600.2.j.e		24
80.s	even	4	1	inner	400.2.s.e	yes	24
80.t	odd	4	1		1600.2.s.e		24

    

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

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 \(S_{2}^{\mathrm{new}}(400, [\chi])\).