Embedded newform 378.2.bf.a.101.9

• Label: 378.2.bf.a.101.9
• 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			101.9
Character	\(\chi\)	\(=\)	378.101
Dual form			378.2.bf.a.131.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.642788 + 0.766044i) q^{2} +(0.930642 + 1.46079i) q^{3} +(-0.173648 - 0.984808i) q^{4} +(-0.879322 + 0.737839i) q^{5} +(-1.71723 - 0.226064i) q^{6} +(2.51837 + 0.811070i) q^{7} +(0.866025 + 0.500000i) q^{8} +(-1.26781 + 2.71894i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

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
							\(3\)	0.930642	+	1.46079i	0.537307	+	0.843387i
\(5\)	−0.879322	+	0.737839i	−0.393245	+	0.329972i								−0.817876	−	0.575395i	\(-0.804848\pi\)
							0.424631	+	0.905367i	\(0.360404\pi\)
\(7\)	2.51837	+	0.811070i	0.951853	+	0.306556i
							\(11\)	−2.40198	+	2.86257i	−0.724226	+	0.863099i	−0.995034	−	0.0995347i	\(-0.968265\pi\)
0.270808	+	0.962633i	\(0.412709\pi\)
\(13\)	0.794026	−	2.18157i	0.220223	−	0.605058i	−0.779550	−	0.626340i	\(-0.784552\pi\)
							0.999774	+	0.0212813i	\(0.00677457\pi\)
\(17\)	1.44778			0.351138			0.175569	−	0.984467i	\(-0.443823\pi\)
							0.175569	+	0.984467i	\(0.443823\pi\)
\(19\)			3.07284i			0.704958i	0.935820	+	0.352479i	\(0.114661\pi\)
							−0.935820	+	0.352479i	\(0.885339\pi\)
\(23\)	−0.580160	+	1.59398i	−0.120972	+	0.332367i	−0.985367	−	0.170446i	\(-0.945479\pi\)
							0.864395	+	0.502813i	\(0.167701\pi\)
\(29\)	−1.08620	−	2.98431i	−0.201702	−	0.554173i	0.797061	−	0.603899i	\(-0.206387\pi\)
							−0.998763	+	0.0497268i	\(0.984165\pi\)
\(31\)	−1.31485	+	0.231844i	−0.236154	+	0.0416404i	−0.290473	−	0.956883i	\(-0.593813\pi\)
							0.0543184	+	0.998524i	\(0.482701\pi\)
\(37\)	4.40928	−	7.63710i	0.724882	−	1.25553i	−0.234141	−	0.972203i	\(-0.575228\pi\)
							0.959023	−	0.283329i	\(-0.0914389\pi\)
\(41\)	−3.11397	−	1.13339i	−0.486321	−	0.177006i	0.0872104	−	0.996190i	\(-0.472205\pi\)
							−0.573532	+	0.819184i	\(0.694427\pi\)
\(43\)	1.16149	−	6.58715i	0.177126	−	1.00453i	−0.758536	−	0.651631i	\(-0.774085\pi\)
							0.935662	−	0.352899i	\(-0.114804\pi\)
\(47\)	−1.49503	+	8.47873i	−0.218072	+	1.23675i	0.657423	+	0.753522i	\(0.271647\pi\)
							−0.875495	+	0.483227i	\(0.839464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.101.9	yes	144
7.5	odd	6		378.2.ba.a.47.24	✓	144
27.23	odd	18		378.2.ba.a.185.24	yes	144
189.131	even	18	inner	378.2.bf.a.131.9	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.47.24	✓	144	7.5	odd	6
378.2.ba.a.185.24	yes	144	27.23	odd	18
378.2.bf.a.101.9	yes	144	1.1	even	1	trivial
378.2.bf.a.131.9	yes	144	189.131	even	18	inner