Newform orbit 222.2.g.a

• Label: 222.2.g.a
• Level: $222$
• Weight: $2$
• Character orbit: 222.g
• Analytic conductor: $1.773$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	222.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.77267892487\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
179.1	−0.707107	+	0.707107i	−1.68174	+	0.414412i		−	1.00000i	−1.01787	−	1.01787i	0.896139	−	1.48221i	−0.256761			0.707107	+	0.707107i	2.65653	−	1.39387i	1.43949
179.2	−0.707107	+	0.707107i	−1.16395	−	1.28266i		−	1.00000i	−0.266957	−	0.266957i	1.73002	+	0.0839450i	−1.78781			0.707107	+	0.707107i	−0.290452	+	2.98591i	0.377534
179.3	−0.707107	+	0.707107i	−0.794453	+	1.53910i		−	1.00000i	1.84813	+	1.84813i	−0.526548	−	1.65007i	5.06606			0.707107	+	0.707107i	−1.73769	−	2.44549i	−2.61365
179.4	−0.707107	+	0.707107i	0.133125	+	1.72693i		−	1.00000i	−2.21702	−	2.21702i	−1.31526	−	1.12699i	−1.69179			0.707107	+	0.707107i	−2.96456	+	0.459796i	3.13535
179.5	−0.707107	+	0.707107i	0.665646	−	1.59904i		−	1.00000i	1.53047	+	1.53047i	0.660007	+	1.60137i	1.09805			0.707107	+	0.707107i	−2.11383	−	2.12878i	−2.16441
179.6	−0.707107	+	0.707107i	1.11385	+	1.32640i		−	1.00000i	2.42933	+	2.42933i	−1.72552	−	0.150296i	−3.86510			0.707107	+	0.707107i	−0.518676	+	2.95482i	−3.43559
179.7	−0.707107	+	0.707107i	1.72752	−	0.125144i		−	1.00000i	−2.30607	−	2.30607i	−1.13305	+	1.31003i	1.43734			0.707107	+	0.707107i	2.96868	−	0.432380i	3.26127
179.8	0.707107	−	0.707107i	−1.72752	−	0.125144i		−	1.00000i	2.30607	+	2.30607i	−1.31003	+	1.13305i	1.43734			−0.707107	−	0.707107i	2.96868	+	0.432380i	3.26127
179.9	0.707107	−	0.707107i	−1.11385	+	1.32640i		−	1.00000i	−2.42933	−	2.42933i	0.150296	+	1.72552i	−3.86510			−0.707107	−	0.707107i	−0.518676	−	2.95482i	−3.43559
179.10	0.707107	−	0.707107i	−0.665646	−	1.59904i		−	1.00000i	−1.53047	−	1.53047i	−1.60137	−	0.660007i	1.09805			−0.707107	−	0.707107i	−2.11383	+	2.12878i	−2.16441
179.11	0.707107	−	0.707107i	−0.133125	+	1.72693i		−	1.00000i	2.21702	+	2.21702i	1.12699	+	1.31526i	−1.69179			−0.707107	−	0.707107i	−2.96456	−	0.459796i	3.13535
179.12	0.707107	−	0.707107i	0.794453	+	1.53910i		−	1.00000i	−1.84813	−	1.84813i	1.65007	+	0.526548i	5.06606			−0.707107	−	0.707107i	−1.73769	+	2.44549i	−2.61365
179.13	0.707107	−	0.707107i	1.16395	−	1.28266i		−	1.00000i	0.266957	+	0.266957i	−0.0839450	−	1.73002i	−1.78781			−0.707107	−	0.707107i	−0.290452	−	2.98591i	0.377534
179.14	0.707107	−	0.707107i	1.68174	+	0.414412i		−	1.00000i	1.01787	+	1.01787i	1.48221	−	0.896139i	−0.256761			−0.707107	−	0.707107i	2.65653	+	1.39387i	1.43949
191.1	−0.707107	−	0.707107i	−1.68174	−	0.414412i			1.00000i	−1.01787	+	1.01787i	0.896139	+	1.48221i	−0.256761			0.707107	−	0.707107i	2.65653	+	1.39387i	1.43949
191.2	−0.707107	−	0.707107i	−1.16395	+	1.28266i			1.00000i	−0.266957	+	0.266957i	1.73002	−	0.0839450i	−1.78781			0.707107	−	0.707107i	−0.290452	−	2.98591i	0.377534
191.3	−0.707107	−	0.707107i	−0.794453	−	1.53910i			1.00000i	1.84813	−	1.84813i	−0.526548	+	1.65007i	5.06606			0.707107	−	0.707107i	−1.73769	+	2.44549i	−2.61365
191.4	−0.707107	−	0.707107i	0.133125	−	1.72693i			1.00000i	−2.21702	+	2.21702i	−1.31526	+	1.12699i	−1.69179			0.707107	−	0.707107i	−2.96456	−	0.459796i	3.13535
191.5	−0.707107	−	0.707107i	0.665646	+	1.59904i			1.00000i	1.53047	−	1.53047i	0.660007	−	1.60137i	1.09805			0.707107	−	0.707107i	−2.11383	+	2.12878i	−2.16441
191.6	−0.707107	−	0.707107i	1.11385	−	1.32640i			1.00000i	2.42933	−	2.42933i	−1.72552	+	0.150296i	−3.86510			0.707107	−	0.707107i	−0.518676	−	2.95482i	−3.43559

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
37.d	odd	4	1	inner
111.g	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	222.2.g.a	✓	28
3.b	odd	2	1	inner	222.2.g.a	✓	28
37.d	odd	4	1	inner	222.2.g.a	✓	28
111.g	even	4	1	inner	222.2.g.a	✓	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
222.2.g.a	✓	28	1.a	even	1	1	trivial
222.2.g.a	✓	28	3.b	odd	2	1	inner
222.2.g.a	✓	28	37.d	odd	4	1	inner
222.2.g.a	✓	28	111.g	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(222, [\chi])\).