Newform orbit 232.2.o.a

• Label: 232.2.o.a
• Level: $232$
• Weight: $2$
• Character orbit: 232.o
• Analytic conductor: $1.853$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 232 = 2^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	232.o (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.85252932689\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(28\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 7 q^{2} - 3 q^{4} - 7 q^{6} - 6 q^{7} - 28 q^{8} - 34 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.41420	−	0.00649190i	−0.00502031	−	0.0219954i	1.99992	+	0.0183617i	−1.75803	+	1.40199i	0.00695692	+	0.0311385i	−0.929975	−	4.07449i	−2.82816	−	0.0389503i	2.70245	−	1.30143i	2.49531	−	1.97127i
5.2	−1.41132	+	0.0903709i	0.298511	+	1.30786i	1.98367	−	0.255085i	0.926834	−	0.739125i	−0.539489	−	1.81884i	0.890488	+	3.90148i	−2.77654	+	0.539273i	1.08151	−	0.520828i	−1.24127	+	1.12690i
5.3	−1.40437	−	0.166569i	−0.718143	−	3.14639i	1.94451	+	0.467848i	−2.33587	+	1.86280i	0.484448	+	4.53831i	0.812981	+	3.56190i	−2.65288	−	0.980927i	−6.68112	+	3.21746i	3.59072	−	2.22697i
5.4	−1.35993	+	0.388047i	−0.490071	−	2.14714i	1.69884	−	1.05544i	3.36609	−	2.68437i	1.49965	+	2.72980i	−0.221909	−	0.972247i	−1.90075	+	2.09455i	−1.66713	+	0.802849i	−3.53600	+	4.95677i
5.5	−1.21267	−	0.727612i	0.556823	+	2.43960i	0.941160	+	1.76471i	2.92235	−	2.33049i	1.09984	−	3.36359i	−0.858332	−	3.76060i	0.142707	−	2.82482i	−2.93869	+	1.41520i	−5.23955	+	0.699795i
5.6	−1.19155	−	0.761711i	0.407988	+	1.78751i	0.839594	+	1.81524i	−1.67377	+	1.33478i	0.875428	−	2.44068i	0.267830	+	1.17344i	0.382265	−	2.80248i	−0.325838	+	0.156915i	3.01110	−	0.315539i
5.7	−1.15074	+	0.822071i	0.327398	+	1.43443i	0.648399	−	1.89198i	0.518163	−	0.413222i	−1.55595	−	1.38150i	0.187302	+	0.820624i	0.809201	+	2.71020i	0.752520	−	0.362395i	−0.256573	+	0.901477i
5.8	−1.05752	+	0.938959i	−0.327398	−	1.43443i	0.236711	−	1.98594i	−0.518163	+	0.413222i	1.69310	+	1.20952i	0.187302	+	0.820624i	1.61439	+	2.32244i	0.752520	−	0.362395i	0.159972	−	0.923526i
5.9	−0.993964	−	1.00600i	−0.476366	−	2.08710i	−0.0240694	+	1.99986i	0.795624	−	0.634489i	−1.62613	+	2.55372i	−0.527958	−	2.31314i	2.03578	−	1.96357i	−1.42614	+	0.686792i	−1.42912	−	0.169738i
5.10	−0.686400	−	1.23647i	−0.112913	−	0.494705i	−1.05771	+	1.69742i	−2.56605	+	2.04636i	−0.534184	+	0.479179i	0.161595	+	0.707996i	2.82482	+	0.142717i	2.47092	−	1.18993i	4.29159	+	1.76822i
5.11	−0.680932	+	1.23949i	0.490071	+	2.14714i	−1.07266	−	1.68801i	−3.36609	+	2.68437i	−2.99506	−	0.854618i	−0.221909	−	0.972247i	2.82269	−	0.180132i	−1.66713	+	0.802849i	−1.03517	−	6.00011i
5.12	−0.402154	+	1.35583i	−0.298511	−	1.30786i	−1.67654	−	1.09050i	−0.926834	+	0.739125i	1.89329	+	0.121232i	0.890488	+	3.90148i	2.15277	−	1.83456i	1.08151	−	0.520828i	−0.629398	−	1.55387i
5.13	−0.401628	−	1.35598i	0.122494	+	0.536679i	−1.67739	+	1.08920i	2.19535	−	1.75073i	0.678532	−	0.381645i	0.552887	+	2.42236i	2.15063	+	1.83706i	2.42989	−	1.17017i	−3.25568	−	2.27371i
5.14	−0.308360	+	1.38019i	0.00502031	+	0.0219954i	−1.80983	−	0.851188i	1.75803	−	1.40199i	−0.0319058		0.000146464i	−0.929975	−	4.07449i	1.73288	−	2.23543i	2.70245	−	1.30143i	1.39289	+	2.85873i
5.15	−0.150109	+	1.40622i	0.718143	+	3.14639i	−1.95493	−	0.422174i	2.33587	−	1.86280i	−4.53233	+	0.537568i	0.812981	+	3.56190i	0.887126	−	2.68570i	−6.68112	+	3.21746i	2.26888	+	3.56439i
5.16	−0.0379686	−	1.41370i	0.699711	+	3.06563i	−1.99712	+	0.107353i	−0.764397	+	0.609586i	4.30733	−	1.10558i	−0.0220649	−	0.0966726i	0.227592	+	2.81926i	−6.20561	+	2.98846i	0.890797	+	1.05749i
5.17	0.149366	−	1.40630i	−0.594132	−	2.60306i	−1.95538	−	0.420107i	0.191735	−	0.152904i	−3.74944	+	0.446722i	−0.581028	−	2.54565i	−0.882865	+	2.68711i	−3.72004	+	1.79148i	−0.186391	−	0.292477i
5.18	0.439524	+	1.34418i	−0.556823	−	2.43960i	−1.61364	+	1.18160i	−2.92235	+	2.33049i	3.03452	−	1.82073i	−0.858332	−	3.76060i	−2.29751	−	1.64968i	−2.93869	+	1.41520i	−4.41704	−	2.90385i
5.19	0.477468	+	1.33117i	−0.407988	−	1.78751i	−1.54405	+	1.27119i	1.67377	−	1.33478i	2.18469	−	1.39658i	0.267830	+	1.17344i	−2.42940	−	1.44845i	−0.325838	+	0.156915i	2.57600	+	1.59076i
5.20	0.661389	−	1.25003i	0.150182	+	0.657992i	−1.12513	−	1.65351i	0.796833	−	0.635453i	0.921836	+	0.247457i	−0.396141	−	1.73561i	−2.81107	+	0.312829i	2.29251	−	1.10401i	−0.267316	−	1.41634i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
29.e	even	14	1	inner
232.o	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	232.2.o.a	✓	168
4.b	odd	2	1		928.2.be.a		168
8.b	even	2	1	inner	232.2.o.a	✓	168
8.d	odd	2	1		928.2.be.a		168
29.e	even	14	1	inner	232.2.o.a	✓	168
116.h	odd	14	1		928.2.be.a		168
232.o	even	14	1	inner	232.2.o.a	✓	168
232.t	odd	14	1		928.2.be.a		168

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
232.2.o.a	✓	168	1.a	even	1	1	trivial
232.2.o.a	✓	168	8.b	even	2	1	inner
232.2.o.a	✓	168	29.e	even	14	1	inner
232.2.o.a	✓	168	232.o	even	14	1	inner
928.2.be.a		168	4.b	odd	2	1
928.2.be.a		168	8.d	odd	2	1
928.2.be.a		168	116.h	odd	14	1
928.2.be.a		168	232.t	odd	14	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(232, [\chi])\).