Newform orbit 500.4.i.a

• Label: 500.4.i.a
• Level: $500$
• Weight: $4$
• Character orbit: 500.i
• Analytic conductor: $29.501$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	500.i (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(29.5009550029\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 122 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
49.1		0		−8.21445	+	2.66904i		0			0			0				24.1794i		0		38.5100	−	27.9792i		0
49.2		0		−8.14260	+	2.64569i		0			0			0			−	36.0715i		0		37.4588	−	27.2154i		0
49.3		0		−3.55368	+	1.15466i		0			0			0				30.1378i		0		−10.5481	+	7.66362i		0
49.4		0		−2.14386	+	0.696582i		0			0			0			−	16.0147i		0		−17.7325	+	12.8835i		0
49.5		0		0.776927	−	0.252439i		0			0			0			−	1.56595i		0		−21.3036	+	15.4779i		0
49.6		0		4.71349	−	1.53150i		0			0			0				16.8592i		0		−1.97201	+	1.43275i		0
49.7		0		5.01325	−	1.62890i		0			0			0			−	9.22290i		0		0.635846	−	0.461969i		0
49.8		0		9.31487	−	3.02658i		0			0			0				0.760275i		0		55.7631	−	40.5142i		0
149.1		0		−4.25695	+	5.85918i		0			0			0				4.97648i		0		−7.86498	−	24.2059i		0
149.2		0		−4.08679	+	5.62498i		0			0			0				19.2903i		0		−6.59510	−	20.2976i		0
149.3		0		−2.66799	+	3.67217i		0			0			0			−	36.0979i		0		1.97677	+	6.08388i		0
149.4		0		0.618939	−	0.851897i		0			0			0				15.9472i		0		8.00082	+	24.6240i		0
149.5		0		0.991787	−	1.36508i		0			0			0			−	19.2807i		0		7.46366	+	22.9708i		0
149.6		0		1.72442	−	2.37346i		0			0			0				7.35306i		0		5.68377	+	17.4928i		0
149.7		0		4.13050	−	5.68515i		0			0			0				21.6979i		0		−6.91642	−	21.2866i		0
149.8		0		5.78214	−	7.95843i		0			0			0			−	24.7439i		0		−21.5601	−	66.3550i		0
349.1		0		−4.25695	−	5.85918i		0			0			0			−	4.97648i		0		−7.86498	+	24.2059i		0
349.2		0		−4.08679	−	5.62498i		0			0			0			−	19.2903i		0		−6.59510	+	20.2976i		0
349.3		0		−2.66799	−	3.67217i		0			0			0				36.0979i		0		1.97677	−	6.08388i		0
349.4		0		0.618939	+	0.851897i		0			0			0			−	15.9472i		0		8.00082	−	24.6240i		0

Inner twists

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

    

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