Newform orbit 168.4.u.a

• Label: 168.4.u.a
• Level: $168$
• Weight: $4$
• Character orbit: 168.u
• Analytic conductor: $9.912$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	168.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.91232088096\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{7} + 14 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0		−5.16688	−	0.550751i		0		−0.746155	−	1.29238i		0		15.9029	+	9.49205i		0		26.3933	+	5.69133i		0
17.2		0		−5.02065	+	1.33907i		0		−2.40532	−	4.16613i		0		−3.40309	+	18.2049i		0		23.4138	−	13.4460i		0
17.3		0		−4.81910	+	1.94327i		0		−9.08545	−	15.7365i		0		−18.4348	+	1.77707i		0		19.4474	−	18.7297i		0
17.4		0		−4.69293	−	2.23079i		0		5.57507	+	9.65631i		0		−7.78587	−	16.8042i		0		17.0472	+	20.9379i		0
17.5		0		−4.65546	+	2.30797i		0		7.91382	+	13.7071i		0		13.3812	−	12.8040i		0		16.3465	−	21.4893i		0
17.6		0		−4.27838	−	2.94880i		0		−5.57507	−	9.65631i		0		−7.78587	−	16.8042i		0		9.60915	+	25.2322i		0
17.7		0		−3.06041	−	4.19928i		0		0.746155	+	1.29238i		0		15.9029	+	9.49205i		0		−8.26784	+	25.7030i		0
17.8		0		−2.82323	+	4.36227i		0		9.22876	+	15.9847i		0		−6.70316	+	17.2646i		0		−11.0587	−	24.6314i		0
17.9		0		−1.88786	+	4.84107i		0		−3.20701	−	5.55470i		0		14.8103	−	11.1200i		0		−19.8720	−	18.2785i		0
17.10		0		−1.40845	+	5.00163i		0		−0.263312	−	0.456070i		0		−16.6747	−	8.05944i		0		−23.0325	−	14.0891i		0
17.11		0		−1.35065	−	5.01754i		0		2.40532	+	4.16613i		0		−3.40309	+	18.2049i		0		−23.3515	+	13.5539i		0
17.12		0		−0.726622	−	5.14510i		0		9.08545	+	15.7365i		0		−18.4348	+	1.77707i		0		−25.9440	+	7.47709i		0
17.13		0		−0.328966	−	5.18573i		0		−7.91382	−	13.7071i		0		13.3812	−	12.8040i		0		−26.7836	+	3.41186i		0
17.14		0		0.919514	+	5.11415i		0		−6.04693	−	10.4736i		0		15.9632	+	9.39028i		0		−25.3090	+	9.40505i		0
17.15		0		2.36622	−	4.62612i		0		−9.22876	−	15.9847i		0		−6.70316	+	17.2646i		0		−15.8020	−	21.8928i		0
17.16		0		2.83134	+	4.35701i		0		−2.97936	−	5.16040i		0		−17.8140	+	5.06584i		0		−10.9670	+	24.6724i		0
17.17		0		3.24856	−	4.05547i		0		3.20701	+	5.55470i		0		14.8103	−	11.1200i		0		−5.89369	−	26.3489i		0
17.18		0		3.29442	+	4.01831i		0		8.29692	+	14.3707i		0		9.36640	+	15.9772i		0		−5.29359	+	26.4760i		0
17.19		0		3.62731	−	3.72057i		0		0.263312	+	0.456070i		0		−16.6747	−	8.05944i		0		−0.685212	−	26.9913i		0
17.20		0		3.72903	+	3.61861i		0		6.11124	+	10.5850i		0		4.39164	−	17.9920i		0		0.811307	+	26.9878i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	168.4.u.a	✓	48
3.b	odd	2	1	inner	168.4.u.a	✓	48
4.b	odd	2	1		336.4.bc.f		48
7.d	odd	6	1	inner	168.4.u.a	✓	48
12.b	even	2	1		336.4.bc.f		48
21.g	even	6	1	inner	168.4.u.a	✓	48
28.f	even	6	1		336.4.bc.f		48
84.j	odd	6	1		336.4.bc.f		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.u.a	✓	48	1.a	even	1	1	trivial
168.4.u.a	✓	48	3.b	odd	2	1	inner
168.4.u.a	✓	48	7.d	odd	6	1	inner
168.4.u.a	✓	48	21.g	even	6	1	inner
336.4.bc.f		48	4.b	odd	2	1
336.4.bc.f		48	12.b	even	2	1
336.4.bc.f		48	28.f	even	6	1
336.4.bc.f		48	84.j	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(168, [\chi])\).