Newform orbit 222.3.p.a

• Label: 222.3.p.a
• Level: $222$
• Weight: $3$
• Character orbit: 222.p
• Analytic conductor: $6.049$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	222.p (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.04906186880\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(26\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 6 q^{3} - 30 q^{7} + 30 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−0.483690	+	1.32893i	−2.93948	+	0.599572i	−1.53209	−	1.28558i	−1.94776	−	0.343443i	0.625007	−	4.19635i	0.406926	−	2.30779i	2.44949	−	1.41421i	8.28103	−	3.52485i	1.39852	−	2.42231i
53.2	−0.483690	+	1.32893i	−2.89274	+	0.795037i	−1.53209	−	1.28558i	4.93658	+	0.870452i	0.342641	−	4.22878i	−0.785889	+	4.45700i	2.44949	−	1.41421i	7.73583	−	4.59966i	−3.54454	+	6.13932i
53.3	−0.483690	+	1.32893i	−2.41837	−	1.77525i	−1.53209	−	1.28558i	1.76236	+	0.310752i	3.52891	−	2.35516i	−0.519179	+	2.94441i	2.44949	−	1.41421i	2.69699	+	8.58640i	−1.26540	+	2.19174i
53.4	−0.483690	+	1.32893i	−2.13867	−	2.10383i	−1.53209	−	1.28558i	−8.76919	−	1.54624i	3.83029	−	1.82453i	0.317942	−	1.80314i	2.44949	−	1.41421i	0.147800	+	8.99879i	6.29641	−	10.9057i
53.5	−0.483690	+	1.32893i	−1.46535	+	2.61777i	−1.53209	−	1.28558i	2.55796	+	0.451038i	−2.77005	−	3.21354i	2.13266	−	12.0949i	2.44949	−	1.41421i	−4.70548	−	7.67192i	−1.83665	+	3.18118i
53.6	−0.483690	+	1.32893i	−1.36273	+	2.67263i	−1.53209	−	1.28558i	−6.11034	−	1.07742i	−2.89260	−	3.10369i	−2.13658	+	12.1171i	2.44949	−	1.41421i	−5.28595	−	7.28414i	4.38732	−	7.59906i
53.7	−0.483690	+	1.32893i	0.231580	−	2.99105i	−1.53209	−	1.28558i	4.95353	+	0.873440i	3.86287	+	1.75449i	−1.92868	+	10.9381i	2.44949	−	1.41421i	−8.89274	−	1.38534i	−3.55671	+	6.16040i
53.8	−0.483690	+	1.32893i	0.843982	−	2.87884i	−1.53209	−	1.28558i	−0.628161	−	0.110762i	3.41753	+	2.51405i	1.23074	−	6.97988i	2.44949	−	1.41421i	−7.57539	−	4.85937i	0.451029	−	0.781205i
53.9	−0.483690	+	1.32893i	1.49480	+	2.60107i	−1.53209	−	1.28558i	4.96727	+	0.875864i	−4.17965	+	0.728366i	−0.784993	+	4.45192i	2.44949	−	1.41421i	−4.53115	+	7.77616i	−3.56658	+	6.17749i
53.10	−0.483690	+	1.32893i	1.70896	+	2.46565i	−1.53209	−	1.28558i	−6.16626	−	1.08728i	−4.10328	+	1.07848i	1.15379	−	6.54347i	2.44949	−	1.41421i	−3.15889	+	8.42742i	4.42747	−	7.66860i
53.11	−0.483690	+	1.32893i	2.78917	−	1.10479i	−1.53209	−	1.28558i	−5.51679	−	0.972759i	0.119090	+	4.24097i	−2.12050	+	12.0260i	2.44949	−	1.41421i	6.55889	−	6.16287i	3.96114	−	6.86089i
53.12	−0.483690	+	1.32893i	2.83024	−	0.994853i	−1.53209	−	1.28558i	7.95245	+	1.40223i	−0.0468717	+	4.24238i	0.636744	−	3.61116i	2.44949	−	1.41421i	7.02053	−	5.63135i	−5.70998	+	9.88997i
53.13	−0.483690	+	1.32893i	2.99224	+	0.215643i	−1.53209	−	1.28558i	−3.22660	−	0.568936i	−1.73389	+	3.87216i	1.35766	−	7.69969i	2.44949	−	1.41421i	8.90700	+	1.29051i	2.31675	−	4.01272i
53.14	0.483690	−	1.32893i	−2.80520	−	1.06342i	−1.53209	−	1.28558i	−2.55796	−	0.451038i	−2.77005	+	3.21354i	2.13266	−	12.0949i	−2.44949	+	1.41421i	6.73827	+	5.96621i	−1.83665	+	3.18118i
53.15	0.483690	−	1.32893i	−2.76185	−	1.17141i	−1.53209	−	1.28558i	6.11034	+	1.07742i	−2.89260	+	3.10369i	−2.13658	+	12.1171i	−2.44949	+	1.41421i	6.25558	+	6.47052i	4.38732	−	7.59906i
53.16	0.483690	−	1.32893i	−2.72700	+	1.25038i	−1.53209	−	1.28558i	−4.93658	−	0.870452i	0.342641	+	4.22878i	−0.785889	+	4.45700i	−2.44949	+	1.41421i	5.87310	−	6.81959i	−3.54454	+	6.13932i
53.17	0.483690	−	1.32893i	−2.63717	+	1.43016i	−1.53209	−	1.28558i	1.94776	+	0.343443i	0.625007	+	4.19635i	0.406926	−	2.30779i	−2.44949	+	1.41421i	4.90929	−	7.54314i	1.39852	−	2.42231i
53.18	0.483690	−	1.32893i	−0.711468	+	2.91441i	−1.53209	−	1.28558i	−1.76236	−	0.310752i	3.52891	+	2.35516i	−0.519179	+	2.94441i	−2.44949	+	1.41421i	−7.98763	−	4.14703i	−1.26540	+	2.19174i
53.19	0.483690	−	1.32893i	−0.526855	−	2.95338i	−1.53209	−	1.28558i	−4.96727	−	0.875864i	−4.17965	−	0.728366i	−0.784993	+	4.45192i	−2.44949	+	1.41421i	−8.44485	+	3.11200i	−3.56658	+	6.17749i
53.20	0.483690	−	1.32893i	−0.285999	+	2.98634i	−1.53209	−	1.28558i	8.76919	+	1.54624i	3.83029	+	1.82453i	0.317942	−	1.80314i	−2.44949	+	1.41421i	−8.83641	−	1.70818i	6.29641	−	10.9057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
37.f	even	9	1	inner
111.p	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	222.3.p.a	✓	156
3.b	odd	2	1	inner	222.3.p.a	✓	156
37.f	even	9	1	inner	222.3.p.a	✓	156
111.p	odd	18	1	inner	222.3.p.a	✓	156

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
222.3.p.a	✓	156	1.a	even	1	1	trivial
222.3.p.a	✓	156	3.b	odd	2	1	inner
222.3.p.a	✓	156	37.f	even	9	1	inner
222.3.p.a	✓	156	111.p	odd	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(222, [\chi])\).