Embedded newform 392.2.z.a.261.10

• Label: 392.2.z.a.261.10
• Level: $392$
• Weight: $2$
• Character: 392.261
• 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			261.10
Character	\(\chi\)	\(=\)	392.261
Dual form			392.2.z.a.389.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19581 + 0.755003i) q^{2} +(0.912188 - 0.0683590i) q^{3} +(0.859941 - 1.80569i) q^{4} +(1.70867 + 2.50615i) q^{5} +(-1.03920 + 0.770449i) q^{6} +(-2.63734 - 0.210865i) q^{7} +(0.334970 + 2.80852i) q^{8} +(-2.13908 + 0.322414i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19581	+	0.755003i	−0.845568	+	0.533868i
							\(3\)	0.912188	−	0.0683590i	0.526652	−	0.0394671i	0.191248	−	0.981542i	\(-0.438746\pi\)
0.335404	+	0.942075i	\(0.391127\pi\)
\(5\)	1.70867	+	2.50615i	0.764139	+	1.12079i	0.989219	+	0.146445i	\(0.0467833\pi\)
							−0.225080	+	0.974340i	\(0.572264\pi\)
\(7\)	−2.63734	−	0.210865i	−0.996819	−	0.0796994i
							\(11\)	−0.926074	+	6.14410i	−0.279222	+	1.85252i	0.206556	+	0.978435i	\(0.433774\pi\)
−0.485778	+	0.874082i	\(0.661464\pi\)
\(13\)	−3.58405	−	2.85819i	−0.994038	−	0.792719i	−0.0157301	−	0.999876i	\(-0.505007\pi\)
							−0.978308	+	0.207158i	\(0.933579\pi\)
\(17\)	−1.96421	+	0.605878i	−0.476390	+	0.146947i	−0.523646	−	0.851936i	\(-0.675429\pi\)
							0.0472558	+	0.998883i	\(0.484952\pi\)
\(19\)	6.08149	+	3.51115i	1.39519	+	0.805513i	0.993884	−	0.110431i	\(-0.0352231\pi\)
							0.401306	+	0.915944i	\(0.368556\pi\)
\(23\)	6.93357	+	2.13872i	1.44575	+	0.445955i	0.915648	−	0.401980i	\(-0.131678\pi\)
							0.530101	+	0.847935i	\(0.322154\pi\)
\(29\)	−5.68081	−	1.29661i	−1.05490	−	0.240774i	−0.340307	−	0.940314i	\(-0.610531\pi\)
							−0.714593	+	0.699540i	\(0.753388\pi\)
\(31\)	0.241848	+	0.418894i	0.0434372	+	0.0752355i	0.886927	−	0.461910i	\(-0.152836\pi\)
							−0.843489	+	0.537146i	\(0.819502\pi\)
\(37\)	2.89927	−	3.12467i	0.476636	−	0.513692i	−0.448264	−	0.893901i	\(-0.647958\pi\)
							0.924900	+	0.380209i	\(0.124148\pi\)
\(41\)	3.03051	+	1.45942i	0.473287	+	0.227923i	0.655294	−	0.755374i	\(-0.272545\pi\)
							−0.182007	+	0.983297i	\(0.558259\pi\)
\(43\)	2.23575	+	4.64258i	0.340949	+	0.707988i	0.998987	−	0.0449969i	\(-0.0143278\pi\)
							−0.658038	+	0.752984i	\(0.728614\pi\)
\(47\)	−1.05759	−	2.69469i	−0.154265	−	0.393061i	0.832758	−	0.553637i	\(-0.186760\pi\)
							−0.987024	+	0.160575i	\(0.948665\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.261.10	✓	648
8.5	even	2	inner	392.2.z.a.261.50	yes	648
49.46	even	21	inner	392.2.z.a.389.50	yes	648
392.389	even	42	inner	392.2.z.a.389.10	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.261.10	✓	648	1.1	even	1	trivial
392.2.z.a.261.50	yes	648	8.5	even	2	inner
392.2.z.a.389.10	yes	648	392.389	even	42	inner
392.2.z.a.389.50	yes	648	49.46	even	21	inner