Newform orbit 1860.2.i.b

• Label: 1860.2.i.b
• Level: $1860$
• Weight: $2$
• Character orbit: 1860.i
• Analytic conductor: $14.852$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1860.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.8521747760\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9} - 8 q^{19} + 64 q^{25} + 24 q^{31} + 8 q^{39} - 16 q^{45} - 160 q^{49} + 68 q^{51} - 24 q^{69} + 52 q^{81}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
929.1		0		−1.72938	−	0.0961633i		0		−1.38738	+	1.75362i		0				3.39946i		0		2.98151	+	0.332606i		0
929.2		0		−1.72938	−	0.0961633i		0		1.38738	+	1.75362i		0			−	3.39946i		0		2.98151	+	0.332606i		0
929.3		0		−1.72938	+	0.0961633i		0		−1.38738	−	1.75362i		0			−	3.39946i		0		2.98151	−	0.332606i		0
929.4		0		−1.72938	+	0.0961633i		0		1.38738	−	1.75362i		0				3.39946i		0		2.98151	−	0.332606i		0
929.5		0		−1.62488	−	0.599792i		0		−2.11867	+	0.714996i		0			−	1.63784i		0		2.28050	+	1.94918i		0
929.6		0		−1.62488	−	0.599792i		0		2.11867	+	0.714996i		0				1.63784i		0		2.28050	+	1.94918i		0
929.7		0		−1.62488	+	0.599792i		0		−2.11867	−	0.714996i		0				1.63784i		0		2.28050	−	1.94918i		0
929.8		0		−1.62488	+	0.599792i		0		2.11867	−	0.714996i		0			−	1.63784i		0		2.28050	−	1.94918i		0
929.9		0		−1.37690	−	1.05079i		0		−2.02481	−	0.948759i		0				3.69636i		0		0.791696	+	2.89365i		0
929.10		0		−1.37690	−	1.05079i		0		2.02481	−	0.948759i		0			−	3.69636i		0		0.791696	+	2.89365i		0
929.11		0		−1.37690	+	1.05079i		0		−2.02481	+	0.948759i		0			−	3.69636i		0		0.791696	−	2.89365i		0
929.12		0		−1.37690	+	1.05079i		0		2.02481	+	0.948759i		0				3.69636i		0		0.791696	−	2.89365i		0
929.13		0		−0.910753	−	1.47327i		0		−1.00177	+	1.99911i		0			−	4.87923i		0		−1.34106	+	2.68357i		0
929.14		0		−0.910753	−	1.47327i		0		1.00177	+	1.99911i		0				4.87923i		0		−1.34106	+	2.68357i		0
929.15		0		−0.910753	+	1.47327i		0		−1.00177	−	1.99911i		0				4.87923i		0		−1.34106	−	2.68357i		0
929.16		0		−0.910753	+	1.47327i		0		1.00177	−	1.99911i		0			−	4.87923i		0		−1.34106	−	2.68357i		0
929.17		0		−0.577293	−	1.63301i		0		−2.10556	−	0.752745i		0			−	0.552709i		0		−2.33347	+	1.88545i		0
929.18		0		−0.577293	−	1.63301i		0		2.10556	−	0.752745i		0				0.552709i		0		−2.33347	+	1.88545i		0
929.19		0		−0.577293	+	1.63301i		0		−2.10556	+	0.752745i		0				0.552709i		0		−2.33347	−	1.88545i		0
929.20		0		−0.577293	+	1.63301i		0		2.10556	+	0.752745i		0			−	0.552709i		0		−2.33347	−	1.88545i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner
31.b	odd	2	1	inner
93.c	even	2	1	inner
155.c	odd	2	1	inner
465.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1860.2.i.b	✓	48
3.b	odd	2	1	inner	1860.2.i.b	✓	48
5.b	even	2	1	inner	1860.2.i.b	✓	48
15.d	odd	2	1	inner	1860.2.i.b	✓	48
31.b	odd	2	1	inner	1860.2.i.b	✓	48
93.c	even	2	1	inner	1860.2.i.b	✓	48
155.c	odd	2	1	inner	1860.2.i.b	✓	48
465.g	even	2	1	inner	1860.2.i.b	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1860.2.i.b	✓	48	1.a	even	1	1	trivial
1860.2.i.b	✓	48	3.b	odd	2	1	inner
1860.2.i.b	✓	48	5.b	even	2	1	inner
1860.2.i.b	✓	48	15.d	odd	2	1	inner
1860.2.i.b	✓	48	31.b	odd	2	1	inner
1860.2.i.b	✓	48	93.c	even	2	1	inner
1860.2.i.b	✓	48	155.c	odd	2	1	inner
1860.2.i.b	✓	48	465.g	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 62T_{7}^{10} + 1425T_{7}^{8} + 15108T_{7}^{6} + 72416T_{7}^{4} + 121444T_{7}^{2} + 30760 \) acting on \(S_{2}^{\mathrm{new}}(1860, [\chi])\).