Newform orbit 342.3.k.a

• Label: 342.3.k.a
• Level: $342$
• Weight: $3$
• Character orbit: 342.k
• Analytic conductor: $9.319$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	342.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.31882504112\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 160 q^{4} + 8 q^{6} + 2 q^{7} + 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
103.1		−	1.41421i	−2.99531	+	0.167628i	−2.00000			−1.81848	−	3.14971i	0.237062	+	4.23601i	4.47452	+	7.75009i			2.82843i	8.94380	−	1.00420i	−4.45436	+	2.57173i
103.2		−	1.41421i	−2.96145	+	0.479394i	−2.00000			−3.12122	−	5.40612i	0.677966	+	4.18812i	−2.64291	−	4.57765i			2.82843i	8.54036	−	2.83940i	−7.64540	+	4.41408i
103.3		−	1.41421i	−2.77255	−	1.14585i	−2.00000			3.21909	+	5.57562i	−1.62047	+	3.92098i	−0.523464	−	0.906666i			2.82843i	6.37406	+	6.35384i	7.88512	−	4.55248i
103.4		−	1.41421i	−2.56150	+	1.56164i	−2.00000			4.44948	+	7.70672i	2.20850	+	3.62250i	−3.65406	−	6.32902i			2.82843i	4.12253	−	8.00029i	10.8989	−	6.29251i
103.5		−	1.41421i	−2.33173	−	1.88760i	−2.00000			0.828174	+	1.43444i	−2.66947	+	3.29756i	−0.842406	−	1.45909i			2.82843i	1.87393	+	8.80275i	2.02860	−	1.17122i
103.6		−	1.41421i	−2.14181	+	2.10063i	−2.00000			0.0905920	+	0.156910i	2.97074	+	3.02898i	−1.74194	−	3.01712i			2.82843i	0.174727	−	8.99830i	0.221904	−	0.128116i
103.7		−	1.41421i	−1.62932	−	2.51899i	−2.00000			−2.02995	−	3.51598i	−3.56239	+	2.30420i	6.66984	+	11.5525i			2.82843i	−3.69065	+	8.20848i	−4.97235	+	2.87079i
103.8		−	1.41421i	−1.04795	−	2.81102i	−2.00000			−4.43902	−	7.68861i	−3.97538	+	1.48202i	−3.42795	−	5.93739i			2.82843i	−6.80361	+	5.89159i	−10.8733	+	6.27773i
103.9		−	1.41421i	−0.795760	+	2.89254i	−2.00000			2.89130	+	5.00788i	4.09066	+	1.12538i	6.30689	+	10.9239i			2.82843i	−7.73353	−	4.60353i	7.08222	−	4.08892i
103.10		−	1.41421i	−0.291452	+	2.98581i	−2.00000			−0.677445	−	1.17337i	4.22257	+	0.412176i	−0.538935	−	0.933463i			2.82843i	−8.83011	−	1.74044i	−1.65939	+	0.958052i
103.11		−	1.41421i	0.237294	−	2.99060i	−2.00000			4.09775	+	7.09751i	−4.22935	−	0.335584i	0.626068	+	1.08438i			2.82843i	−8.88738	−	1.41930i	10.0374	−	5.79510i
103.12		−	1.41421i	1.11386	+	2.78556i	−2.00000			0.401271	+	0.695022i	3.93937	−	1.57524i	−6.36922	−	11.0318i			2.82843i	−6.51863	+	6.20543i	0.982909	−	0.567483i
103.13		−	1.41421i	1.16014	−	2.76660i	−2.00000			−0.338479	−	0.586263i	−3.91256	−	1.64068i	−3.06441	−	5.30771i			2.82843i	−6.30816	−	6.41928i	−0.829101	+	0.478682i
103.14		−	1.41421i	1.27828	+	2.71404i	−2.00000			−2.20717	−	3.82292i	3.83823	−	1.80776i	0.576502	+	0.998530i			2.82843i	−5.73201	+	6.93859i	−5.40643	+	3.12140i
103.15		−	1.41421i	1.83089	−	2.37652i	−2.00000			1.41190	+	2.44548i	−3.36090	−	2.58927i	4.65995	+	8.07128i			2.82843i	−2.29568	−	8.70229i	3.45843	−	1.99673i
103.16		−	1.41421i	2.47249	+	1.69905i	−2.00000			−3.91366	−	6.77866i	2.40282	−	3.49663i	2.90904	+	5.03860i			2.82843i	3.22645	+	8.40179i	−9.58648	+	5.53476i
103.17		−	1.41421i	2.77377	−	1.14289i	−2.00000			−3.55515	−	6.15769i	−1.61629	−	3.92270i	−2.42656	−	4.20293i			2.82843i	6.38761	−	6.34023i	−8.70829	+	5.02773i
103.18		−	1.41421i	2.78112	+	1.12489i	−2.00000			2.17195	+	3.76193i	1.59083	−	3.93310i	1.73122	+	2.99856i			2.82843i	6.46924	+	6.25691i	5.32017	−	3.07160i
103.19		−	1.41421i	2.93853	−	0.604170i	−2.00000			3.04086	+	5.26692i	−0.854425	−	4.15571i	−6.65082	−	11.5196i			2.82843i	8.26996	−	3.55075i	7.44854	−	4.30042i
103.20		−	1.41421i	2.94245	−	0.584777i	−2.00000			−0.501779	−	0.869107i	−0.827000	−	4.16126i	4.42865	+	7.67065i			2.82843i	8.31607	−	3.44136i	−1.22910	+	0.709623i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	342.3.k.a	✓	80
3.b	odd	2	1		1026.3.k.a		80
9.c	even	3	1		342.3.t.a	yes	80
9.d	odd	6	1		1026.3.t.a		80
19.d	odd	6	1		342.3.t.a	yes	80
57.f	even	6	1		1026.3.t.a		80
171.i	odd	6	1	inner	342.3.k.a	✓	80
171.t	even	6	1		1026.3.k.a		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
342.3.k.a	✓	80	1.a	even	1	1	trivial
342.3.k.a	✓	80	171.i	odd	6	1	inner
342.3.t.a	yes	80	9.c	even	3	1
342.3.t.a	yes	80	19.d	odd	6	1
1026.3.k.a		80	3.b	odd	2	1
1026.3.k.a		80	171.t	even	6	1
1026.3.t.a		80	9.d	odd	6	1
1026.3.t.a		80	57.f	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(342, [\chi])\).