Embedded newform 392.2.z.a.109.38

• Label: 392.2.z.a.109.38
• Level: $392$
• Weight: $2$
• Character: 392.109
• 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			109.38
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.751927 + 1.19775i) q^{2} +(-0.115819 - 0.768410i) q^{3} +(-0.869210 + 1.80124i) q^{4} +(3.94542 + 1.54846i) q^{5} +(0.833275 - 0.716511i) q^{6} +(-2.02706 + 1.70031i) q^{7} +(-2.81102 + 0.313307i) q^{8} +(2.28968 - 0.706272i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.751927	+	1.19775i	0.531693	+	0.846937i
							\(3\)	−0.115819	−	0.768410i	−0.0668682	−	0.443642i	−0.997064	−	0.0765770i	\(-0.975601\pi\)
0.930195	−	0.367065i	\(-0.119637\pi\)
\(5\)	3.94542	+	1.54846i	1.76445	+	0.692495i	0.998786	+	0.0492500i	\(0.0156831\pi\)
							0.765661	+	0.643244i	\(0.222412\pi\)
\(7\)	−2.02706	+	1.70031i	−0.766156	+	0.642655i
							\(11\)	0.256000	−	0.829933i	0.0771870	−	0.250234i	−0.908753	−	0.417335i	\(-0.862964\pi\)
0.985940	+	0.167101i	\(0.0534404\pi\)
\(13\)	−3.90995	+	0.892420i	−1.08442	+	0.247513i	−0.727144	−	0.686484i	\(-0.759153\pi\)
							−0.357280	+	0.933997i	\(0.616296\pi\)
\(17\)	4.54072	−	3.09581i	1.10129	−	0.750845i	0.130659	−	0.991427i	\(-0.458291\pi\)
							0.970629	+	0.240582i	\(0.0773384\pi\)
\(19\)	−2.86428	+	1.65369i	−0.657112	+	0.379384i	−0.791176	−	0.611589i	\(-0.790531\pi\)
							0.134064	+	0.990973i	\(0.457197\pi\)
\(23\)	2.63259	+	1.79487i	0.548932	+	0.374255i	0.805797	−	0.592192i	\(-0.201737\pi\)
							−0.256865	+	0.966447i	\(0.582690\pi\)
\(29\)	0.706191	−	1.46642i	0.131136	−	0.272307i	−0.825054	−	0.565055i	\(-0.808855\pi\)
							0.956190	+	0.292747i	\(0.0945694\pi\)
\(31\)	−2.09297	+	3.62514i	−0.375909	+	0.651094i	−0.990463	−	0.137781i	\(-0.956003\pi\)
							0.614553	+	0.788875i	\(0.289336\pi\)
\(37\)	−6.13653	+	0.459869i	−1.00884	+	0.0756020i	−0.568884	−	0.822418i	\(-0.692625\pi\)
							−0.439955	+	0.898020i	\(0.645006\pi\)
\(41\)	−5.02276	−	6.29834i	−0.784423	−	0.983636i	−0.999974	−	0.00714301i	\(-0.997726\pi\)
							0.215551	−	0.976493i	\(-0.430845\pi\)
\(43\)	2.78552	+	2.22137i	0.424787	+	0.338757i	0.812436	−	0.583051i	\(-0.198141\pi\)
							−0.387649	+	0.921807i	\(0.626713\pi\)
\(47\)	9.02623	−	8.37512i	1.31661	−	1.22164i	0.359079	−	0.933307i	\(-0.383091\pi\)
							0.957531	−	0.288329i	\(-0.0930997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.38	yes	648
8.5	even	2	inner	392.2.z.a.109.13	✓	648
49.9	even	21	inner	392.2.z.a.205.13	yes	648
392.205	even	42	inner	392.2.z.a.205.38	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.13	✓	648	8.5	even	2	inner
392.2.z.a.109.38	yes	648	1.1	even	1	trivial
392.2.z.a.205.13	yes	648	49.9	even	21	inner
392.2.z.a.205.38	yes	648	392.205	even	42	inner