Newform orbit 3300.2.x.d

• Label: 3300.2.x.d
• Level: $3300$
• Weight: $2$
• Character orbit: 3300.x
• Analytic conductor: $26.351$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.3506326670\)
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{11} + 16 q^{31} - 32 q^{71} - 32 q^{81} + 80 q^{91}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
1693.1		0		−0.707107	−	0.707107i		0			0			0		−3.37614	−	3.37614i		0				1.00000i		0
1693.2		0		−0.707107	−	0.707107i		0			0			0		3.37614	+	3.37614i		0				1.00000i		0
1693.3		0		−0.707107	−	0.707107i		0			0			0		−0.256514	−	0.256514i		0				1.00000i		0
1693.4		0		−0.707107	−	0.707107i		0			0			0		0.256514	+	0.256514i		0				1.00000i		0
1693.5		0		−0.707107	−	0.707107i		0			0			0		−2.08903	−	2.08903i		0				1.00000i		0
1693.6		0		−0.707107	−	0.707107i		0			0			0		2.08903	+	2.08903i		0				1.00000i		0
1693.7		0		−0.707107	−	0.707107i		0			0			0		−0.414559	−	0.414559i		0				1.00000i		0
1693.8		0		−0.707107	−	0.707107i		0			0			0		0.414559	+	0.414559i		0				1.00000i		0
1693.9		0		0.707107	+	0.707107i		0			0			0		−2.08903	−	2.08903i		0				1.00000i		0
1693.10		0		0.707107	+	0.707107i		0			0			0		2.08903	+	2.08903i		0				1.00000i		0
1693.11		0		0.707107	+	0.707107i		0			0			0		−0.414559	−	0.414559i		0				1.00000i		0
1693.12		0		0.707107	+	0.707107i		0			0			0		0.414559	+	0.414559i		0				1.00000i		0
1693.13		0		0.707107	+	0.707107i		0			0			0		−3.37614	−	3.37614i		0				1.00000i		0
1693.14		0		0.707107	+	0.707107i		0			0			0		3.37614	+	3.37614i		0				1.00000i		0
1693.15		0		0.707107	+	0.707107i		0			0			0		−0.256514	−	0.256514i		0				1.00000i		0
1693.16		0		0.707107	+	0.707107i		0			0			0		0.256514	+	0.256514i		0				1.00000i		0
1957.1		0		−0.707107	+	0.707107i		0			0			0		−3.37614	+	3.37614i		0			−	1.00000i		0
1957.2		0		−0.707107	+	0.707107i		0			0			0		3.37614	−	3.37614i		0			−	1.00000i		0
1957.3		0		−0.707107	+	0.707107i		0			0			0		−0.256514	+	0.256514i		0			−	1.00000i		0
1957.4		0		−0.707107	+	0.707107i		0			0			0		0.256514	−	0.256514i		0			−	1.00000i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner
11.b	odd	2	1	inner
55.d	odd	2	1	inner
55.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3300.2.x.d	✓	32
5.b	even	2	1	inner	3300.2.x.d	✓	32
5.c	odd	4	2	inner	3300.2.x.d	✓	32
11.b	odd	2	1	inner	3300.2.x.d	✓	32
55.d	odd	2	1	inner	3300.2.x.d	✓	32
55.e	even	4	2	inner	3300.2.x.d	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3300.2.x.d	✓	32	1.a	even	1	1	trivial
3300.2.x.d	✓	32	5.b	even	2	1	inner
3300.2.x.d	✓	32	5.c	odd	4	2	inner
3300.2.x.d	✓	32	11.b	odd	2	1	inner
3300.2.x.d	✓	32	55.d	odd	2	1	inner
3300.2.x.d	✓	32	55.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3300, [\chi])\):

\( T_{7}^{16} + 596T_{7}^{12} + 39670T_{7}^{8} + 5364T_{7}^{4} + 81 \)

\( T_{19}^{8} - 88T_{19}^{6} + 2046T_{19}^{4} - 14632T_{19}^{2} + 20449 \)