Embedded newform 378.2.bf.a.101.18

• Label: 378.2.bf.a.101.18
• Level: $378$
• Weight: $2$
• Character: 378.101
• Analytic conductor: $3.018$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.bf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.01834519640\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.18
Character	\(\chi\)	\(=\)	378.101
Dual form			378.2.bf.a.131.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642788 - 0.766044i) q^{2} +(0.327343 + 1.70084i) q^{3} +(-0.173648 - 0.984808i) q^{4} +(1.33249 - 1.11809i) q^{5} +(1.51333 + 0.842517i) q^{6} +(1.31779 - 2.29421i) q^{7} +(-0.866025 - 0.500000i) q^{8} +(-2.78569 + 1.11351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.642788	−	0.766044i	0.454519	−	0.541675i
							\(3\)	0.327343	+	1.70084i	0.188992	+	0.981979i
\(5\)	1.33249	−	1.11809i	0.595907	−	0.500026i								−0.294220	−	0.955738i	\(-0.595060\pi\)
							0.890127	+	0.455712i	\(0.150615\pi\)
\(7\)	1.31779	−	2.29421i	0.498079	−	0.867131i
							\(11\)	1.64951	−	1.96581i	0.497347	−	0.592715i	−0.457723	−	0.889095i	\(-0.651335\pi\)
0.955070	+	0.296379i	\(0.0957793\pi\)
\(13\)	0.456584	−	1.25445i	0.126634	−	0.347923i	−0.860133	−	0.510070i	\(-0.829620\pi\)
							0.986767	+	0.162147i	\(0.0518418\pi\)
\(17\)	3.74095			0.907313			0.453657	−	0.891177i	\(-0.350119\pi\)
							0.453657	+	0.891177i	\(0.350119\pi\)
\(19\)			5.43587i			1.24707i	0.781794	+	0.623537i	\(0.214305\pi\)
							−0.781794	+	0.623537i	\(0.785695\pi\)
\(23\)	−1.55735	+	4.27877i	−0.324729	+	0.892186i	0.664692	+	0.747117i	\(0.268563\pi\)
							−0.989422	+	0.145069i	\(0.953660\pi\)
\(29\)	−1.79379	−	4.92840i	−0.333099	−	0.915181i	−0.987301	−	0.158861i	\(-0.949218\pi\)
							0.654202	−	0.756320i	\(-0.273004\pi\)
\(31\)	3.35018	−	0.590727i	0.601710	−	0.106098i	0.135508	−	0.990776i	\(-0.456733\pi\)
							0.466201	+	0.884679i	\(0.345622\pi\)
\(37\)	−1.39821	+	2.42177i	−0.229865	+	0.398137i	−0.957768	−	0.287543i	\(-0.907162\pi\)
							0.727903	+	0.685680i	\(0.240495\pi\)
\(41\)	−5.82495	−	2.12011i	−0.909704	−	0.331105i	−0.155569	−	0.987825i	\(-0.549721\pi\)
							−0.754135	+	0.656720i	\(0.771943\pi\)
\(43\)	−1.74954	+	9.92212i	−0.266802	+	1.51311i	0.497052	+	0.867721i	\(0.334416\pi\)
							−0.763854	+	0.645389i	\(0.776695\pi\)
\(47\)	1.51762	−	8.60687i	0.221368	−	1.25544i	−0.648139	−	0.761522i	\(-0.724453\pi\)
							0.869507	−	0.493920i	\(-0.164436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.101.18	yes	144
7.5	odd	6		378.2.ba.a.47.11	✓	144
27.23	odd	18		378.2.ba.a.185.11	yes	144
189.131	even	18	inner	378.2.bf.a.131.18	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.47.11	✓	144	7.5	odd	6
378.2.ba.a.185.11	yes	144	27.23	odd	18
378.2.bf.a.101.18	yes	144	1.1	even	1	trivial
378.2.bf.a.131.18	yes	144	189.131	even	18	inner