Newform orbit 500.4.g.b

• Label: 500.4.g.b
• Level: $500$
• Weight: $4$
• Character orbit: 500.g
• Analytic conductor: $29.501$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	500.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(29.5009550029\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{5})\)
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 244 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
101.1		0		−7.95843	−	5.78214i		0			0			0		24.7439				0		21.5601	+	66.3550i		0
101.2		0		−5.85918	−	4.25695i		0			0			0		4.97648				0		7.86498	+	24.2059i		0
101.3		0		−5.68515	−	4.13050i		0			0			0		−21.6979				0		6.91642	+	21.2866i		0
101.4		0		−5.62498	−	4.08679i		0			0			0		19.2903				0		6.59510	+	20.2976i		0
101.5		0		−3.67217	−	2.66799i		0			0			0		−36.0979				0		−1.97677	−	6.08388i		0
101.6		0		−2.37346	−	1.72442i		0			0			0		−7.35306				0		−5.68377	−	17.4928i		0
101.7		0		−1.36508	−	0.991787i		0			0			0		19.2807				0		−7.46366	−	22.9708i		0
101.8		0		−0.851897	−	0.618939i		0			0			0		−15.9472				0		−8.00082	−	24.6240i		0
101.9		0		0.851897	+	0.618939i		0			0			0		15.9472				0		−8.00082	−	24.6240i		0
101.10		0		1.36508	+	0.991787i		0			0			0		−19.2807				0		−7.46366	−	22.9708i		0
101.11		0		2.37346	+	1.72442i		0			0			0		7.35306				0		−5.68377	−	17.4928i		0
101.12		0		3.67217	+	2.66799i		0			0			0		36.0979				0		−1.97677	−	6.08388i		0
101.13		0		5.62498	+	4.08679i		0			0			0		−19.2903				0		6.59510	+	20.2976i		0
101.14		0		5.68515	+	4.13050i		0			0			0		21.6979				0		6.91642	+	21.2866i		0
101.15		0		5.85918	+	4.25695i		0			0			0		−4.97648				0		7.86498	+	24.2059i		0
101.16		0		7.95843	+	5.78214i		0			0			0		−24.7439				0		21.5601	+	66.3550i		0
201.1		0		−3.02658	−	9.31487i		0			0			0		−0.760275				0		−55.7631	+	40.5142i		0
201.2		0		−2.66904	−	8.21445i		0			0			0		24.1794				0		−38.5100	+	27.9792i		0
201.3		0		−2.64569	−	8.14260i		0			0			0		−36.0715				0		−37.4588	+	27.2154i		0
201.4		0		−1.62890	−	5.01325i		0			0			0		9.22290				0		−0.635846	+	0.461969i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
25.d	even	5	1	inner
25.e	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	500.4.g.b		64
5.b	even	2	1	inner	500.4.g.b		64
5.c	odd	4	1		100.4.i.a	✓	32
5.c	odd	4	1		500.4.i.a		32
25.d	even	5	1	inner	500.4.g.b		64
25.d	even	5	1		2500.4.a.g		32
25.e	even	10	1	inner	500.4.g.b		64
25.e	even	10	1		2500.4.a.g		32
25.f	odd	20	1		100.4.i.a	✓	32
25.f	odd	20	1		500.4.i.a		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
100.4.i.a	✓	32	5.c	odd	4	1
100.4.i.a	✓	32	25.f	odd	20	1
500.4.g.b		64	1.a	even	1	1	trivial
500.4.g.b		64	5.b	even	2	1	inner
500.4.g.b		64	25.d	even	5	1	inner
500.4.g.b		64	25.e	even	10	1	inner
500.4.i.a		32	5.c	odd	4	1
500.4.i.a		32	25.f	odd	20	1
2500.4.a.g		32	25.d	even	5	1
2500.4.a.g		32	25.e	even	10	1