Newform orbit 4004.2.e.a

• Label: 4004.2.e.a
• Level: $4004$
• Weight: $2$
• Character orbit: 4004.e
• Analytic conductor: $31.972$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(31.9721009693\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3849.1		0			−	3.23317i		0				0.717019i		0		−0.284426	+	2.63042i		0		−7.45341				0
3849.2		0			−	3.15005i		0			−	2.09918i		0		1.59876	−	2.10807i		0		−6.92283				0
3849.3		0			−	3.04894i		0				4.30537i		0		−2.63608	−	0.225969i		0		−6.29604				0
3849.4		0			−	2.99006i		0				1.25381i		0		−2.36377	−	1.18853i		0		−5.94047				0
3849.5		0			−	2.95250i		0				1.47372i		0		0.700125	+	2.55144i		0		−5.71727				0
3849.6		0			−	2.73693i		0			−	3.14756i		0		−1.11363	−	2.39996i		0		−4.49077				0
3849.7		0			−	2.42199i		0			−	0.343428i		0		2.23070	−	1.42266i		0		−2.86603				0
3849.8		0			−	2.41288i		0				2.25637i		0		2.16066	−	1.52694i		0		−2.82198				0
3849.9		0			−	2.39724i		0			−	2.83291i		0		2.55736	+	0.678178i		0		−2.74675				0
3849.10		0			−	2.30818i		0				0.586386i		0		−2.63094	−	0.279585i		0		−2.32771				0
3849.11		0			−	1.96820i		0			−	3.88595i		0		−0.0756292	+	2.64467i		0		−0.873803				0
3849.12		0			−	1.77770i		0				3.16364i		0		0.435138	+	2.60972i		0		−0.160220				0
3849.13		0			−	1.76248i		0			−	1.09765i		0		−1.82169	+	1.91871i		0		−0.106338				0
3849.14		0			−	1.33626i		0				2.57665i		0		2.48110	+	0.918779i		0		1.21441				0
3849.15		0			−	1.12498i		0			−	0.542613i		0		1.59076	−	2.11411i		0		1.73443				0
3849.16		0			−	1.04079i		0			−	4.22530i		0		−2.02914	+	1.69782i		0		1.91676				0
3849.17		0			−	1.00653i		0				1.31275i		0		−1.40205	−	2.24371i		0		1.98690				0
3849.18		0			−	0.968020i		0			−	0.815183i		0		−2.53164	+	0.768635i		0		2.06294				0
3849.19		0			−	0.950260i		0				3.52969i		0		−2.60374	+	0.469588i		0		2.09701				0
3849.20		0			−	0.867297i		0			−	0.687824i		0		2.28648	+	1.33116i		0		2.24780				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
77.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4004.2.e.a	✓	48
7.b	odd	2	1		4004.2.e.b	yes	48
11.b	odd	2	1		4004.2.e.b	yes	48
77.b	even	2	1	inner	4004.2.e.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4004.2.e.a	✓	48	1.a	even	1	1	trivial
4004.2.e.a	✓	48	77.b	even	2	1	inner
4004.2.e.b	yes	48	7.b	odd	2	1
4004.2.e.b	yes	48	11.b	odd	2	1