Newform orbit 252.4.e.a

• Label: 252.4.e.a
• Level: $252$
• Weight: $4$
• Character orbit: 252.e
• Analytic conductor: $14.868$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.8684813214\)
Analytic rank:	\(0\)
Dimension:	\(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 24 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
71.1	−2.82720	−	0.0833946i		0		7.98609	+	0.471546i		−	8.62666i		0			−	7.00000i	−22.5389	−	1.99915i		0		−0.719417	+	24.3893i
71.2	−2.82720	+	0.0833946i		0		7.98609	−	0.471546i			8.62666i		0				7.00000i	−22.5389	+	1.99915i		0		−0.719417	−	24.3893i
71.3	−2.45159	−	1.41057i		0		4.02056	+	6.91629i			16.9233i		0			−	7.00000i	−0.100815	−	22.6272i		0		23.8716	−	41.4889i
71.4	−2.45159	+	1.41057i		0		4.02056	−	6.91629i		−	16.9233i		0				7.00000i	−0.100815	+	22.6272i		0		23.8716	+	41.4889i
71.5	−2.36083	−	1.55771i		0		3.14707	+	7.35499i		−	8.56325i		0				7.00000i	4.02725	−	22.2661i		0		−13.3391	+	20.2164i
71.6	−2.36083	+	1.55771i		0		3.14707	−	7.35499i			8.56325i		0			−	7.00000i	4.02725	+	22.2661i		0		−13.3391	−	20.2164i
71.7	−2.27330	−	1.68288i		0		2.33582	+	7.65140i			4.43347i		0			−	7.00000i	7.56635	−	21.3249i		0		7.46100	−	10.0786i
71.8	−2.27330	+	1.68288i		0		2.33582	−	7.65140i		−	4.43347i		0				7.00000i	7.56635	+	21.3249i		0		7.46100	+	10.0786i
71.9	−1.88294	−	2.11058i		0		−0.909099	+	7.94818i			21.8666i		0				7.00000i	18.4870	−	13.0472i		0		46.1512	−	41.1734i
71.10	−1.88294	+	2.11058i		0		−0.909099	−	7.94818i		−	21.8666i		0			−	7.00000i	18.4870	+	13.0472i		0		46.1512	+	41.1734i
71.11	−1.42938	−	2.44067i		0		−3.91377	+	6.97728i		−	15.7775i		0				7.00000i	22.6235	−	0.420902i		0		−38.5077	+	22.5520i
71.12	−1.42938	+	2.44067i		0		−3.91377	−	6.97728i			15.7775i		0			−	7.00000i	22.6235	+	0.420902i		0		−38.5077	−	22.5520i
71.13	−1.16208	−	2.57868i		0		−5.29916	+	5.99324i		−	0.300247i		0			−	7.00000i	21.6127	+	6.70021i		0		−0.774240	+	0.348910i
71.14	−1.16208	+	2.57868i		0		−5.29916	−	5.99324i			0.300247i		0				7.00000i	21.6127	−	6.70021i		0		−0.774240	−	0.348910i
71.15	−0.968530	−	2.65743i		0		−6.12390	+	5.14761i			12.6551i		0				7.00000i	19.6106	+	11.2882i		0		33.6300	−	12.2568i
71.16	−0.968530	+	2.65743i		0		−6.12390	−	5.14761i		−	12.6551i		0			−	7.00000i	19.6106	−	11.2882i		0		33.6300	+	12.2568i
71.17	−0.614971	−	2.76076i		0		−7.24362	+	3.39558i			2.97985i		0			−	7.00000i	13.8290	+	17.9097i		0		8.22665	−	1.83252i
71.18	−0.614971	+	2.76076i		0		−7.24362	−	3.39558i		−	2.97985i		0				7.00000i	13.8290	−	17.9097i		0		8.22665	+	1.83252i
71.19	0.614971	−	2.76076i		0		−7.24362	−	3.39558i			2.97985i		0				7.00000i	−13.8290	+	17.9097i		0		8.22665	+	1.83252i
71.20	0.614971	+	2.76076i		0		−7.24362	+	3.39558i		−	2.97985i		0			−	7.00000i	−13.8290	−	17.9097i		0		8.22665	−	1.83252i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.4.e.a	✓	36
3.b	odd	2	1	inner	252.4.e.a	✓	36
4.b	odd	2	1	inner	252.4.e.a	✓	36
12.b	even	2	1	inner	252.4.e.a	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.4.e.a	✓	36	1.a	even	1	1	trivial
252.4.e.a	✓	36	3.b	odd	2	1	inner
252.4.e.a	✓	36	4.b	odd	2	1	inner
252.4.e.a	✓	36	12.b	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(252, [\chi])\).