Embedded newform 378.2.bf.a.101.20

• Label: 378.2.bf.a.101.20
• 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			101.20
Character	\(\chi\)	\(=\)	378.101
Dual form			378.2.bf.a.131.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642788 - 0.766044i) q^{2} +(0.520556 - 1.65197i) q^{3} +(-0.173648 - 0.984808i) q^{4} +(1.37026 - 1.14979i) q^{5} +(-0.930879 - 1.46064i) q^{6} +(-1.38267 + 2.25571i) q^{7} +(-0.866025 - 0.500000i) q^{8} +(-2.45804 - 1.71989i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

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
							\(3\)	0.520556	−	1.65197i	0.300543	−	0.953768i
\(5\)	1.37026	−	1.14979i	0.612799	−	0.514200i								−0.282732	−	0.959199i	\(-0.591241\pi\)
							0.895531	+	0.444999i	\(0.146796\pi\)
\(7\)	−1.38267	+	2.25571i	−0.522602	+	0.852577i
							\(11\)	3.15337	−	3.75804i	0.950777	−	1.13309i	−0.0402177	−	0.999191i	\(-0.512805\pi\)
0.990995	−	0.133901i	\(-0.0427504\pi\)
\(13\)	0.0313706	−	0.0861899i	0.00870063	−	0.0239048i	−0.935266	−	0.353945i	\(-0.884840\pi\)
							0.943967	+	0.330040i	\(0.107062\pi\)
\(17\)	−0.123596			−0.0299764			−0.0149882	−	0.999888i	\(-0.504771\pi\)
							−0.0149882	+	0.999888i	\(0.504771\pi\)
\(19\)		−	2.33825i		−	0.536432i	−0.963359	−	0.268216i	\(-0.913566\pi\)
							0.963359	−	0.268216i	\(-0.0864341\pi\)
\(23\)	−1.95762	+	5.37851i	−0.408191	+	1.12150i	0.549949	+	0.835198i	\(0.314647\pi\)
							−0.958141	+	0.286298i	\(0.907575\pi\)
\(29\)	1.25242	+	3.44100i	0.232569	+	0.638978i	0.999998	−	0.00215645i	\(-0.000686419\pi\)
							−0.767429	+	0.641134i	\(0.778464\pi\)
\(31\)	4.32390	−	0.762419i	0.776594	−	0.136935i	0.228717	−	0.973493i	\(-0.426547\pi\)
							0.547878	+	0.836558i	\(0.315436\pi\)
\(37\)	−2.72254	+	4.71557i	−0.447582	+	0.775236i	−0.998228	−	0.0595036i	\(-0.981048\pi\)
							0.550646	+	0.834739i	\(0.314382\pi\)
\(41\)	5.84036	+	2.12572i	0.912111	+	0.331981i	0.755096	−	0.655615i	\(-0.227590\pi\)
							0.157016	+	0.987596i	\(0.449813\pi\)
\(43\)	1.72827	−	9.80152i	0.263559	−	1.49472i	−0.509548	−	0.860442i	\(-0.670188\pi\)
							0.773107	−	0.634275i	\(-0.218701\pi\)
\(47\)	1.03169	−	5.85102i	0.150488	−	0.853459i	−0.812308	−	0.583229i	\(-0.801789\pi\)
							0.962796	−	0.270230i	\(-0.0870997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.101.20	yes	144
7.5	odd	6		378.2.ba.a.47.4	✓	144
27.23	odd	18		378.2.ba.a.185.4	yes	144
189.131	even	18	inner	378.2.bf.a.131.20	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.47.4	✓	144	7.5	odd	6
378.2.ba.a.185.4	yes	144	27.23	odd	18
378.2.bf.a.101.20	yes	144	1.1	even	1	trivial
378.2.bf.a.131.20	yes	144	189.131	even	18	inner