Newform orbit 342.3.l.a

• Label: 342.3.l.a
• Level: $342$
• Weight: $3$
• Character orbit: 342.l
• Analytic conductor: $9.319$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	342.l (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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
151.1	−1.22474	−	0.707107i	−2.99547	−	0.164850i	1.00000	+	1.73205i	−3.44954	−	5.97479i	3.55212	+	2.32001i	3.48951	−	6.04402i		−	2.82843i	8.94565	+	0.987607i			9.75679i
151.2	−1.22474	−	0.707107i	−2.98964	−	0.249157i	1.00000	+	1.73205i	2.82095	+	4.88603i	3.48536	+	2.41915i	0.230492	−	0.399224i		−	2.82843i	8.87584	+	1.48978i		−	7.97885i
151.3	−1.22474	−	0.707107i	−2.70026	+	1.30712i	1.00000	+	1.73205i	2.01557	+	3.49107i	4.23141	+	0.308482i	−5.00320	+	8.66581i		−	2.82843i	5.58285	−	7.05916i		−	5.70089i
151.4	−1.22474	−	0.707107i	−2.56998	−	1.54765i	1.00000	+	1.73205i	0.947945	+	1.64189i	2.05321	+	3.71273i	4.99227	−	8.64687i		−	2.82843i	4.20955	+	7.95486i		−	2.68119i
151.5	−1.22474	−	0.707107i	−2.40146	−	1.79805i	1.00000	+	1.73205i	−0.436276	−	0.755652i	1.66976	+	3.90024i	−5.65898	+	9.80164i		−	2.82843i	2.53402	+	8.63590i			1.23397i
151.6	−1.22474	−	0.707107i	−2.33101	+	1.88849i	1.00000	+	1.73205i	−0.682011	−	1.18128i	4.19026	−	0.664650i	−1.09727	+	1.90053i		−	2.82843i	1.86719	−	8.80418i			1.92902i
151.7	−1.22474	−	0.707107i	−1.69573	+	2.47477i	1.00000	+	1.73205i	4.69486	+	8.13173i	3.82677	−	1.83190i	5.67137	−	9.82311i		−	2.82843i	−3.24900	−	8.39309i		−	13.2791i
151.8	−1.22474	−	0.707107i	−1.23487	−	2.73406i	1.00000	+	1.73205i	3.37473	+	5.84521i	−0.420866	+	4.22171i	0.541915	−	0.938625i		−	2.82843i	−5.95017	+	6.75244i		−	9.54518i
151.9	−1.22474	−	0.707107i	−0.749940	−	2.90475i	1.00000	+	1.73205i	−2.70510	−	4.68538i	−1.13549	+	4.08787i	−0.415272	+	0.719272i		−	2.82843i	−7.87518	+	4.35678i			7.65119i
151.10	−1.22474	−	0.707107i	−0.212313	+	2.99248i	1.00000	+	1.73205i	−4.45237	−	7.71173i	2.37603	−	3.51489i	−6.19455	+	10.7293i		−	2.82843i	−8.90985	−	1.27068i			12.5932i
151.11	−1.22474	−	0.707107i	−0.165289	+	2.99544i	1.00000	+	1.73205i	−0.937637	−	1.62404i	2.32053	−	3.55178i	3.77691	−	6.54179i		−	2.82843i	−8.94536	−	0.990225i			2.65204i
151.12	−1.22474	−	0.707107i	0.436393	−	2.96809i	1.00000	+	1.73205i	−3.91686	−	6.78421i	−2.63323	+	3.32658i	2.89551	−	5.01517i		−	2.82843i	−8.61912	−	2.59051i			11.0786i
151.13	−1.22474	−	0.707107i	0.630103	−	2.93308i	1.00000	+	1.73205i	2.38430	+	4.12974i	−2.84572	+	3.14673i	−1.38550	+	2.39976i		−	2.82843i	−8.20594	−	3.69629i		−	6.74383i
151.14	−1.22474	−	0.707107i	1.11114	+	2.78664i	1.00000	+	1.73205i	1.92987	+	3.34264i	0.609589	−	4.19862i	−2.60142	+	4.50580i		−	2.82843i	−6.53073	+	6.19270i		−	5.45850i
151.15	−1.22474	−	0.707107i	1.96099	−	2.27035i	1.00000	+	1.73205i	−1.90135	−	3.29324i	−4.00710	+	1.39397i	−4.83213	+	8.36949i		−	2.82843i	−1.30902	−	8.90429i			5.37784i
151.16	−1.22474	−	0.707107i	2.06353	+	2.17757i	1.00000	+	1.73205i	0.484620	+	0.839386i	−0.987525	−	4.12611i	1.11147	−	1.92512i		−	2.82843i	−0.483659	+	8.98699i		−	1.37071i
151.17	−1.22474	−	0.707107i	2.61697	−	1.46679i	1.00000	+	1.73205i	0.350777	+	0.607564i	−4.24230	−	0.0540357i	5.86746	−	10.1627i		−	2.82843i	4.69706	−	7.67708i		−	0.992148i
151.18	−1.22474	−	0.707107i	2.89344	+	0.792462i	1.00000	+	1.73205i	−2.91053	−	5.04119i	−2.98337	−	3.01654i	−2.84300	+	4.92423i		−	2.82843i	7.74401	+	4.58588i			8.23222i
151.19	−1.22474	−	0.707107i	2.92156	+	0.681538i	1.00000	+	1.73205i	−2.19924	−	3.80920i	−3.09624	−	2.90056i	2.29487	−	3.97483i		−	2.82843i	8.07101	+	3.98231i			6.22040i
151.20	−1.22474	−	0.707107i	2.96233	−	0.473898i	1.00000	+	1.73205i	4.58731	+	7.94545i	−3.96320	−	1.51428i	−1.84045	+	3.18775i		−	2.82843i	8.55084	−	2.80769i		−	12.9749i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
19.b	odd	2	1	inner
171.o	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.l.a	✓	80
3.b	odd	2	1		1026.3.l.a		80
9.c	even	3	1	inner	342.3.l.a	✓	80
9.d	odd	6	1		1026.3.l.a		80
19.b	odd	2	1	inner	342.3.l.a	✓	80
57.d	even	2	1		1026.3.l.a		80
171.l	even	6	1		1026.3.l.a		80
171.o	odd	6	1	inner	342.3.l.a	✓	80

    

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

Hecke kernels

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