Embedded newform 378.2.bf.a.5.2

• Label: 378.2.bf.a.5.2
• Level: $378$
• Weight: $2$
• Character: 378.5
• 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			5.2
Character	\(\chi\)	\(=\)	378.5
Dual form			378.2.bf.a.227.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.984808 + 0.173648i) q^{2} +(-1.68531 + 0.399650i) q^{3} +(0.939693 - 0.342020i) q^{4} +(-0.662712 + 3.75843i) q^{5} +(1.59031 - 0.686230i) q^{6} +(-0.925433 + 2.47862i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(2.68056 - 1.34707i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.984808	+	0.173648i	−0.696364	+	0.122788i
							\(3\)	−1.68531	+	0.399650i	−0.973016	+	0.230738i
\(5\)	−0.662712	+	3.75843i	−0.296374	+	1.68082i								0.365192	+	0.930932i	\(0.381003\pi\)
							−0.661566	+	0.749887i	\(0.730108\pi\)
\(7\)	−0.925433	+	2.47862i	−0.349781	+	0.936832i
							\(11\)	−3.19616	+	0.563570i	−0.963679	+	0.169923i	−0.633283	−	0.773920i	\(-0.718293\pi\)
−0.330396	+	0.943843i	\(0.607182\pi\)
\(13\)	1.31704	−	1.56959i	0.365282	−	0.435327i	−0.551829	−	0.833957i	\(-0.686070\pi\)
							0.917112	+	0.398631i	\(0.130514\pi\)
\(17\)	3.42321			0.830250			0.415125	−	0.909764i	\(-0.363738\pi\)
							0.415125	+	0.909764i	\(0.363738\pi\)
\(19\)		−	1.62680i		−	0.373214i	−0.982435	−	0.186607i	\(-0.940251\pi\)
							0.982435	−	0.186607i	\(-0.0597491\pi\)
\(23\)	−5.04242	+	6.00932i	−1.05142	+	1.25303i	−0.0849095	+	0.996389i	\(0.527060\pi\)
							−0.966507	+	0.256641i	\(0.917384\pi\)
\(29\)	−5.87835	−	7.00554i	−1.09158	−	1.30090i	−0.950440	−	0.310907i	\(-0.899367\pi\)
							−0.141141	−	0.989989i	\(-0.545077\pi\)
\(31\)	−1.09293	−	3.00281i	−0.196297	−	0.539320i	0.802022	−	0.597295i	\(-0.203758\pi\)
							−0.998318	+	0.0579749i	\(0.981536\pi\)
\(37\)	−1.01761	−	1.76255i	−0.167293	−	0.289761i	0.770174	−	0.637834i	\(-0.220169\pi\)
							−0.937467	+	0.348073i	\(0.886836\pi\)
\(41\)	2.22692	+	1.86861i	0.347787	+	0.291828i	0.799901	−	0.600133i	\(-0.204886\pi\)
							−0.452114	+	0.891960i	\(0.649330\pi\)
\(43\)	−1.35405	−	0.492834i	−0.206491	−	0.0751564i	0.236704	−	0.971582i	\(-0.423933\pi\)
							−0.443195	+	0.896425i	\(0.646155\pi\)
\(47\)	9.75116	+	3.54913i	1.42235	+	0.517694i	0.934729	−	0.355360i	\(-0.115642\pi\)
							0.487623	+	0.873054i	\(0.337864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.5.2	yes	144
7.3	odd	6		378.2.ba.a.59.15	✓	144
27.11	odd	18		378.2.ba.a.173.15	yes	144
189.38	even	18	inner	378.2.bf.a.227.2	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.59.15	✓	144	7.3	odd	6
378.2.ba.a.173.15	yes	144	27.11	odd	18
378.2.bf.a.5.2	yes	144	1.1	even	1	trivial
378.2.bf.a.227.2	yes	144	189.38	even	18	inner