Newform orbit 1428.2.g.a

• Label: 1428.2.g.a
• Level: $1428$
• Weight: $2$
• Character orbit: 1428.g
• Analytic conductor: $11.403$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1428.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.4026374086\)
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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
713.1		0		−1.72009	−	0.203175i		0			−	3.53682i		0		−1.39891	+	2.24568i		0		2.91744	+	0.698961i		0
713.2		0		−1.72009	−	0.203175i		0			−	3.53682i		0		−1.39891	−	2.24568i		0		2.91744	+	0.698961i		0
713.3		0		−1.72009	+	0.203175i		0				3.53682i		0		−1.39891	−	2.24568i		0		2.91744	−	0.698961i		0
713.4		0		−1.72009	+	0.203175i		0				3.53682i		0		−1.39891	+	2.24568i		0		2.91744	−	0.698961i		0
713.5		0		−1.55620	−	0.760421i		0			−	1.05590i		0		0.793083	−	2.52409i		0		1.84352	+	2.36674i		0
713.6		0		−1.55620	−	0.760421i		0			−	1.05590i		0		0.793083	+	2.52409i		0		1.84352	+	2.36674i		0
713.7		0		−1.55620	+	0.760421i		0				1.05590i		0		0.793083	+	2.52409i		0		1.84352	−	2.36674i		0
713.8		0		−1.55620	+	0.760421i		0				1.05590i		0		0.793083	−	2.52409i		0		1.84352	−	2.36674i		0
713.9		0		−1.30066	−	1.14380i		0				3.45888i		0		2.49878	−	0.869546i		0		0.383456	+	2.97539i		0
713.10		0		−1.30066	−	1.14380i		0				3.45888i		0		2.49878	+	0.869546i		0		0.383456	+	2.97539i		0
713.11		0		−1.30066	+	1.14380i		0			−	3.45888i		0		2.49878	+	0.869546i		0		0.383456	−	2.97539i		0
713.12		0		−1.30066	+	1.14380i		0			−	3.45888i		0		2.49878	−	0.869546i		0		0.383456	−	2.97539i		0
713.13		0		−1.29168	−	1.15393i		0				0.939112i		0		−2.45277	−	0.991924i		0		0.336889	+	2.98102i		0
713.14		0		−1.29168	−	1.15393i		0				0.939112i		0		−2.45277	+	0.991924i		0		0.336889	+	2.98102i		0
713.15		0		−1.29168	+	1.15393i		0			−	0.939112i		0		−2.45277	+	0.991924i		0		0.336889	−	2.98102i		0
713.16		0		−1.29168	+	1.15393i		0			−	0.939112i		0		−2.45277	−	0.991924i		0		0.336889	−	2.98102i		0
713.17		0		−0.633738	−	1.61195i		0			−	2.71037i		0		1.90411	−	1.83694i		0		−2.19675	+	2.04311i		0
713.18		0		−0.633738	−	1.61195i		0			−	2.71037i		0		1.90411	+	1.83694i		0		−2.19675	+	2.04311i		0
713.19		0		−0.633738	+	1.61195i		0				2.71037i		0		1.90411	+	1.83694i		0		−2.19675	−	2.04311i		0
713.20		0		−0.633738	+	1.61195i		0				2.71037i		0		1.90411	−	1.83694i		0		−2.19675	−	2.04311i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
17.b	even	2	1	inner
21.c	even	2	1	inner
51.c	odd	2	1	inner
119.d	odd	2	1	inner
357.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1428.2.g.a	✓	48
3.b	odd	2	1	inner	1428.2.g.a	✓	48
7.b	odd	2	1	inner	1428.2.g.a	✓	48
17.b	even	2	1	inner	1428.2.g.a	✓	48
21.c	even	2	1	inner	1428.2.g.a	✓	48
51.c	odd	2	1	inner	1428.2.g.a	✓	48
119.d	odd	2	1	inner	1428.2.g.a	✓	48
357.c	even	2	1	inner	1428.2.g.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1428.2.g.a	✓	48	1.a	even	1	1	trivial
1428.2.g.a	✓	48	3.b	odd	2	1	inner
1428.2.g.a	✓	48	7.b	odd	2	1	inner
1428.2.g.a	✓	48	17.b	even	2	1	inner
1428.2.g.a	✓	48	21.c	even	2	1	inner
1428.2.g.a	✓	48	51.c	odd	2	1	inner
1428.2.g.a	✓	48	119.d	odd	2	1	inner
1428.2.g.a	✓	48	357.c	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1428, [\chi])\).