Newform orbit 1368.2.g.e

• Label: 1368.2.g.e
• Level: $1368$
• Weight: $2$
• Character orbit: 1368.g
• Analytic conductor: $10.924$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9235349965\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{4} - 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
685.1	−1.41062	−	0.100750i		0		1.97970	+	0.284241i		−	1.67412i		0		−1.51494			−2.76397	−	0.600411i		0		−0.168669	+	2.36155i
685.2	−1.41062	+	0.100750i		0		1.97970	−	0.284241i			1.67412i		0		−1.51494			−2.76397	+	0.600411i		0		−0.168669	−	2.36155i
685.3	−1.39336	−	0.241964i		0		1.88291	+	0.674286i		−	2.43078i		0		3.63857			−2.46041	−	1.39512i		0		−0.588160	+	3.38695i
685.4	−1.39336	+	0.241964i		0		1.88291	−	0.674286i			2.43078i		0		3.63857			−2.46041	+	1.39512i		0		−0.588160	−	3.38695i
685.5	−1.21912	−	0.716763i		0		0.972500	+	1.74764i			1.33114i		0		−0.925029			0.0670504	−	2.82763i		0		0.954111	−	1.62282i
685.6	−1.21912	+	0.716763i		0		0.972500	−	1.74764i		−	1.33114i		0		−0.925029			0.0670504	+	2.82763i		0		0.954111	+	1.62282i
685.7	−1.18397	−	0.773451i		0		0.803547	+	1.83148i			2.06615i		0		−3.95479			0.465187	−	2.78991i		0		1.59806	−	2.44625i
685.8	−1.18397	+	0.773451i		0		0.803547	−	1.83148i		−	2.06615i		0		−3.95479			0.465187	+	2.78991i		0		1.59806	+	2.44625i
685.9	−1.11141	−	0.874513i		0		0.470455	+	1.94388i		−	2.65611i		0		1.28466			1.17708	−	2.57186i		0		−2.32280	+	2.95202i
685.10	−1.11141	+	0.874513i		0		0.470455	−	1.94388i			2.65611i		0		1.28466			1.17708	+	2.57186i		0		−2.32280	−	2.95202i
685.11	−0.735397	−	1.20797i		0		−0.918383	+	1.77667i			0.364416i		0		4.21282			2.82155	−	0.197182i		0		0.440203	−	0.267990i
685.12	−0.735397	+	1.20797i		0		−0.918383	−	1.77667i		−	0.364416i		0		4.21282			2.82155	+	0.197182i		0		0.440203	+	0.267990i
685.13	−0.727952	−	1.21247i		0		−0.940171	+	1.76524i		−	3.45115i		0		−4.25855			2.82470	−	0.145082i		0		−4.18442	+	2.51227i
685.14	−0.727952	+	1.21247i		0		−0.940171	−	1.76524i			3.45115i		0		−4.25855			2.82470	+	0.145082i		0		−4.18442	−	2.51227i
685.15	−0.591826	−	1.28442i		0		−1.29948	+	1.52031i			4.21648i		0		0.835621			2.72179	+	0.769327i		0		5.41574	−	2.49542i
685.16	−0.591826	+	1.28442i		0		−1.29948	−	1.52031i		−	4.21648i		0		0.835621			2.72179	−	0.769327i		0		5.41574	+	2.49542i
685.17	−0.156412	−	1.40554i		0		−1.95107	+	0.439685i			0.608970i		0		−1.31836			0.923164	+	2.67353i		0		0.855931	−	0.0952501i
685.18	−0.156412	+	1.40554i		0		−1.95107	−	0.439685i		−	0.608970i		0		−1.31836			0.923164	−	2.67353i		0		0.855931	+	0.0952501i
685.19	0.156412	−	1.40554i		0		−1.95107	−	0.439685i			0.608970i		0		−1.31836			−0.923164	+	2.67353i		0		0.855931	+	0.0952501i
685.20	0.156412	+	1.40554i		0		−1.95107	+	0.439685i		−	0.608970i		0		−1.31836			−0.923164	−	2.67353i		0		0.855931	−	0.0952501i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.b	even	2	1	inner
24.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1368.2.g.e	✓	36
3.b	odd	2	1	inner	1368.2.g.e	✓	36
4.b	odd	2	1		5472.2.g.e		36
8.b	even	2	1	inner	1368.2.g.e	✓	36
8.d	odd	2	1		5472.2.g.e		36
12.b	even	2	1		5472.2.g.e		36
24.f	even	2	1		5472.2.g.e		36
24.h	odd	2	1	inner	1368.2.g.e	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1368.2.g.e	✓	36	1.a	even	1	1	trivial
1368.2.g.e	✓	36	3.b	odd	2	1	inner
1368.2.g.e	✓	36	8.b	even	2	1	inner
1368.2.g.e	✓	36	24.h	odd	2	1	inner
5472.2.g.e		36	4.b	odd	2	1
5472.2.g.e		36	8.d	odd	2	1
5472.2.g.e		36	12.b	even	2	1
5472.2.g.e		36	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^18 + 52*T5^16 + 1066*T5^14 + 11220*T5^12 + 66025*T5^10 + 220192*T5^8 + 398900*T5^6 + 347872*T5^4 + 109056*T5^2 + 9216