Newform orbit 500.2.e.c

• Label: 500.2.e.c
• Level: $500$
• Weight: $2$
• Character orbit: 500.e
• Analytic conductor: $3.993$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	500.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.99252010106\)
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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
307.1	−1.39022	−	0.259395i	0.519784	+	0.519784i	1.86543	+	0.721232i		0		−0.587785	−	0.857444i	−2.15096	+	2.15096i	−2.40627	−	1.48655i		−	2.45965i		0
307.2	−1.38743	+	0.273914i	0.572461	+	0.572461i	1.84994	−	0.760074i		0		−0.951057	−	0.637447i	1.95303	−	1.95303i	−2.35848	+	1.56128i		−	2.34458i		0
307.3	−1.34942	−	0.423159i	−1.02677	−	1.02677i	1.64187	+	1.14204i		0		0.951057	+	1.82003i	−1.08889	+	1.08889i	−1.73231	−	2.23587i		−	0.891491i		0
307.4	−1.14764	−	0.826394i	−1.82972	−	1.82972i	0.634146	+	1.89680i		0		0.587785	+	3.61192i	0.611043	−	0.611043i	0.839736	−	2.70090i			3.69572i		0
307.5	−0.826394	−	1.14764i	1.82972	+	1.82972i	−0.634146	+	1.89680i		0		0.587785	−	3.61192i	−0.611043	+	0.611043i	2.70090	−	0.839736i			3.69572i		0
307.6	−0.423159	−	1.34942i	1.02677	+	1.02677i	−1.64187	+	1.14204i		0		0.951057	−	1.82003i	1.08889	−	1.08889i	2.23587	+	1.73231i		−	0.891491i		0
307.7	−0.273914	+	1.38743i	0.572461	+	0.572461i	−1.84994	−	0.760074i		0		−0.951057	+	0.637447i	1.95303	−	1.95303i	1.56128	−	2.35848i		−	2.34458i		0
307.8	−0.259395	−	1.39022i	−0.519784	−	0.519784i	−1.86543	+	0.721232i		0		−0.587785	+	0.857444i	2.15096	−	2.15096i	1.48655	+	2.40627i		−	2.45965i		0
307.9	0.259395	+	1.39022i	0.519784	+	0.519784i	−1.86543	+	0.721232i		0		−0.587785	+	0.857444i	−2.15096	+	2.15096i	−1.48655	−	2.40627i		−	2.45965i		0
307.10	0.273914	−	1.38743i	−0.572461	−	0.572461i	−1.84994	−	0.760074i		0		−0.951057	+	0.637447i	−1.95303	+	1.95303i	−1.56128	+	2.35848i		−	2.34458i		0
307.11	0.423159	+	1.34942i	−1.02677	−	1.02677i	−1.64187	+	1.14204i		0		0.951057	−	1.82003i	−1.08889	+	1.08889i	−2.23587	−	1.73231i		−	0.891491i		0
307.12	0.826394	+	1.14764i	−1.82972	−	1.82972i	−0.634146	+	1.89680i		0		0.587785	−	3.61192i	0.611043	−	0.611043i	−2.70090	+	0.839736i			3.69572i		0
307.13	1.14764	+	0.826394i	1.82972	+	1.82972i	0.634146	+	1.89680i		0		0.587785	+	3.61192i	−0.611043	+	0.611043i	−0.839736	+	2.70090i			3.69572i		0
307.14	1.34942	+	0.423159i	1.02677	+	1.02677i	1.64187	+	1.14204i		0		0.951057	+	1.82003i	1.08889	−	1.08889i	1.73231	+	2.23587i		−	0.891491i		0
307.15	1.38743	−	0.273914i	−0.572461	−	0.572461i	1.84994	−	0.760074i		0		−0.951057	−	0.637447i	−1.95303	+	1.95303i	2.35848	−	1.56128i		−	2.34458i		0
307.16	1.39022	+	0.259395i	−0.519784	−	0.519784i	1.86543	+	0.721232i		0		−0.587785	−	0.857444i	2.15096	−	2.15096i	2.40627	+	1.48655i		−	2.45965i		0
443.1	−1.39022	+	0.259395i	0.519784	−	0.519784i	1.86543	−	0.721232i		0		−0.587785	+	0.857444i	−2.15096	−	2.15096i	−2.40627	+	1.48655i			2.45965i		0
443.2	−1.38743	−	0.273914i	0.572461	−	0.572461i	1.84994	+	0.760074i		0		−0.951057	+	0.637447i	1.95303	+	1.95303i	−2.35848	−	1.56128i			2.34458i		0
443.3	−1.34942	+	0.423159i	−1.02677	+	1.02677i	1.64187	−	1.14204i		0		0.951057	−	1.82003i	−1.08889	−	1.08889i	−1.73231	+	2.23587i			0.891491i		0
443.4	−1.14764	+	0.826394i	−1.82972	+	1.82972i	0.634146	−	1.89680i		0		0.587785	−	3.61192i	0.611043	+	0.611043i	0.839736	+	2.70090i		−	3.69572i		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	500.2.e.c	✓	32
4.b	odd	2	1	inner	500.2.e.c	✓	32
5.b	even	2	1	inner	500.2.e.c	✓	32
5.c	odd	4	2	inner	500.2.e.c	✓	32
20.d	odd	2	1	inner	500.2.e.c	✓	32
20.e	even	4	2	inner	500.2.e.c	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
500.2.e.c	✓	32	1.a	even	1	1	trivial
500.2.e.c	✓	32	4.b	odd	2	1	inner
500.2.e.c	✓	32	5.b	even	2	1	inner
500.2.e.c	✓	32	5.c	odd	4	2	inner
500.2.e.c	✓	32	20.d	odd	2	1	inner
500.2.e.c	✓	32	20.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 50T_{3}^{12} + 235T_{3}^{8} + 150T_{3}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\).