Newform orbit 507.2.x.a

• Label: 507.2.x.a
• Level: $507$
• Weight: $2$
• Character orbit: 507.x
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $48$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.x (of order \(156\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{156}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1		0		1.38468	−	1.04052i	1.99838	+	0.0805319i		0			0		−1.33289	+	2.01672i		0		0.834652	−	2.88155i		0
11.1		0		1.59345	−	0.678906i	−1.06893	−	1.69038i		0			0		1.78351	−	3.57116i		0		2.07817	−	2.16361i		0
20.1		0		1.19983	−	1.24916i	1.80690	+	0.857385i		0			0		−0.697469	+	4.91487i		0		−0.120798	−	2.99757i		0
32.1		0		1.72643	−	0.139372i	1.95958	+	0.400051i		0			0		−0.421909	−	1.87333i		0		2.96115	−	0.481234i		0
41.1		0		−0.346455	+	1.69705i	−1.92104	−	0.556435i		0			0		−0.766841	−	1.06446i		0		−2.75994	−	1.17590i		0
50.1		0		−1.64291	−	0.548485i	1.44240	+	1.38545i		0			0		0.680973	+	0.0688013i		0		2.39833	+	1.80223i		0
59.1		0		−1.46391	+	0.925722i	0.320823	+	1.97410i		0			0		5.25731	+	0.105888i		0		1.28608	−	2.71035i		0
71.1		0		−1.64291	+	0.548485i	1.44240	−	1.38545i		0			0		0.680973	−	0.0688013i		0		2.39833	−	1.80223i		0
98.1		0		1.64291	−	0.548485i	−1.44240	+	1.38545i		0			0		−0.527444	−	5.22047i		0		2.39833	−	1.80223i		0
110.1		0		1.46391	−	0.925722i	−0.320823	−	1.97410i		0			0		0.0119040	−	0.591030i		0		1.28608	−	2.71035i		0
119.1		0		1.64291	+	0.548485i	−1.44240	−	1.38545i		0			0		−0.527444	+	5.22047i		0		2.39833	+	1.80223i		0
128.1		0		0.346455	−	1.69705i	1.92104	+	0.556435i		0			0		4.15936	−	2.99643i		0		−2.75994	−	1.17590i		0
137.1		0		−1.72643	+	0.139372i	−1.95958	−	0.400051i		0			0		−4.81030	+	1.08337i		0		2.96115	−	0.481234i		0
149.1		0		−1.19983	+	1.24916i	−1.80690	−	0.857385i		0			0		1.81421	+	0.257455i		0		−0.120798	−	2.99757i		0
158.1		0		−1.59345	+	0.678906i	1.06893	+	1.69038i		0			0		3.10761	+	1.55200i		0		2.07817	−	2.16361i		0
167.1		0		−1.38468	+	1.04052i	−1.99838	−	0.0805319i		0			0		3.92688	+	2.59536i		0		0.834652	−	2.88155i		0
176.1		0		1.70962	+	0.277840i	−0.783933	−	1.83996i		0			0		1.45291	+	4.07679i		0		2.84561	+	0.950004i		0
197.1		0		0.742517	−	1.56482i	−0.633336	−	1.89707i		0			0		−0.0345359	+	0.0318602i		0		−1.89734	−	2.32381i		0
206.1		0		1.72643	+	0.139372i	1.95958	−	0.400051i		0			0		−0.421909	+	1.87333i		0		2.96115	+	0.481234i		0
215.1		0		−1.59345	−	0.678906i	1.06893	−	1.69038i		0			0		3.10761	−	1.55200i		0		2.07817	+	2.16361i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
169.l	odd	156	1	inner
507.x	even	156	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	507.2.x.a	✓	48
3.b	odd	2	1	CM	507.2.x.a	✓	48
169.l	odd	156	1	inner	507.2.x.a	✓	48
507.x	even	156	1	inner	507.2.x.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
507.2.x.a	✓	48	1.a	even	1	1	trivial
507.2.x.a	✓	48	3.b	odd	2	1	CM
507.2.x.a	✓	48	169.l	odd	156	1	inner
507.2.x.a	✓	48	507.x	even	156	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\).