Embedded newform 392.2.z.a.221.18

• Label: 392.2.z.a.221.18
• Level: $392$
• Weight: $2$
• Character: 392.221
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			221.18
Character	\(\chi\)	\(=\)	392.221
Dual form			392.2.z.a.149.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.764757 + 1.18960i) q^{2} +(2.31986 - 0.910478i) q^{3} +(-0.830294 - 1.81951i) q^{4} +(0.243207 - 1.61357i) q^{5} +(-0.691025 + 3.45600i) q^{6} +(2.38367 + 1.14809i) q^{7} +(2.79946 + 0.403765i) q^{8} +(2.35363 - 2.18384i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{8}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.764757	+	1.18960i	−0.540765	+	0.841174i
							\(3\)	2.31986	−	0.910478i	1.33937	−	0.525665i	0.415928	−	0.909398i	\(-0.363457\pi\)
0.923444	+	0.383733i	\(0.125362\pi\)
\(5\)	0.243207	−	1.61357i	0.108766	−	0.721612i	−0.866272	−	0.499573i	\(-0.833490\pi\)
							0.975038	−	0.222040i	\(-0.0712715\pi\)
\(7\)	2.38367	+	1.14809i	0.900943	+	0.433938i
							\(11\)	−0.368051	+	0.396664i	−0.110971	+	0.119599i	−0.786094	−	0.618106i	\(-0.787900\pi\)
0.675123	+	0.737705i	\(0.264091\pi\)
\(13\)	−4.52991	−	1.03392i	−1.25637	−	0.286758i	−0.458015	−	0.888945i	\(-0.651439\pi\)
							−0.798355	+	0.602186i	\(0.794296\pi\)
\(17\)	0.0399285	−	0.532809i	0.00968409	−	0.129225i	−0.990266	−	0.139186i	\(-0.955552\pi\)
							0.999950	+	0.00996045i	\(0.00317056\pi\)
\(19\)	7.33373	−	4.23413i	1.68247	−	0.971376i	0.722462	−	0.691411i	\(-0.243011\pi\)
							0.960010	−	0.279965i	\(-0.0903227\pi\)
\(23\)	−0.0838842	−	1.11936i	−0.0174911	−	0.233402i	−0.999097	−	0.0424845i	\(-0.986473\pi\)
							0.981606	−	0.190918i	\(-0.0611464\pi\)
\(29\)	−1.09726	−	2.27848i	−0.203755	−	0.423102i	0.773903	−	0.633305i	\(-0.218302\pi\)
							−0.977658	+	0.210202i	\(0.932588\pi\)
\(31\)	−3.77942	+	6.54615i	−0.678804	+	1.17572i	0.296537	+	0.955021i	\(0.404168\pi\)
							−0.975341	+	0.220702i	\(0.929165\pi\)
\(37\)	2.60422	−	3.81969i	0.428131	−	0.627953i	−0.550192	−	0.835038i	\(-0.685445\pi\)
							0.978323	+	0.207086i	\(0.0663979\pi\)
\(41\)	−3.12394	+	3.91729i	−0.487877	+	0.611778i	−0.963447	−	0.267900i	\(-0.913670\pi\)
							0.475570	+	0.879678i	\(0.342242\pi\)
\(43\)	−0.436876	+	0.348397i	−0.0666229	+	0.0531300i	−0.656235	−	0.754556i	\(-0.727852\pi\)
							0.589612	+	0.807687i	\(0.299281\pi\)
\(47\)	−9.59872	+	2.96081i	−1.40012	+	0.431879i	−0.900684	−	0.434474i	\(-0.856934\pi\)
							−0.499433	+	0.866353i	\(0.666458\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.221.18	yes	648
8.5	even	2	inner	392.2.z.a.221.9	yes	648
49.2	even	21	inner	392.2.z.a.149.9	✓	648
392.149	even	42	inner	392.2.z.a.149.18	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.149.9	✓	648	49.2	even	21	inner
392.2.z.a.149.18	yes	648	392.149	even	42	inner
392.2.z.a.221.9	yes	648	8.5	even	2	inner
392.2.z.a.221.18	yes	648	1.1	even	1	trivial