Embedded newform 392.2.z.a.109.36

• Label: 392.2.z.a.109.36
• 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			109.36
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.750720 + 1.19851i) q^{2} +(0.379898 + 2.52046i) q^{3} +(-0.872840 + 1.79949i) q^{4} +(0.0390419 + 0.0153228i) q^{5} +(-2.73559 + 2.34747i) q^{6} +(-0.309625 + 2.62757i) q^{7} +(-2.81196 + 0.304804i) q^{8} +(-3.34168 + 1.03077i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.750720	+	1.19851i	0.530839	+	0.847473i
							\(3\)	0.379898	+	2.52046i	0.219334	+	1.45519i	0.779289	+	0.626665i	\(0.215580\pi\)
−0.559955	+	0.828523i	\(0.689181\pi\)
\(5\)	0.0390419	+	0.0153228i	0.0174601	+	0.00685257i	0.374055	−	0.927407i	\(-0.377967\pi\)
							−0.356595	+	0.934259i	\(0.616062\pi\)
\(7\)	−0.309625	+	2.62757i	−0.117027	+	0.993129i
							\(11\)	1.55790	−	5.05057i	0.469723	−	1.52281i	−0.342534	−	0.939506i	\(-0.611285\pi\)
0.812257	−	0.583300i	\(-0.198239\pi\)
\(13\)	3.75907	−	0.857983i	1.04258	−	0.237962i	0.333249	−	0.942839i	\(-0.391855\pi\)
							0.709329	+	0.704877i	\(0.248998\pi\)
\(17\)	−3.63112	+	2.47566i	−0.880676	+	0.600435i	−0.916992	−	0.398905i	\(-0.869390\pi\)
							0.0363158	+	0.999340i	\(0.488438\pi\)
\(19\)	5.71005	−	3.29670i	1.30998	−	0.756315i	0.327884	−	0.944718i	\(-0.393664\pi\)
							0.982092	+	0.188403i	\(0.0603310\pi\)
\(23\)	5.76658	+	3.93159i	1.20242	+	0.819793i	0.987620	−	0.156866i	\(-0.0501392\pi\)
							0.214795	+	0.976659i	\(0.431092\pi\)
\(29\)	−3.82589	+	7.94454i	−0.710449	+	1.47526i	0.162119	+	0.986771i	\(0.448167\pi\)
							−0.872568	+	0.488492i	\(0.837547\pi\)
\(31\)	−0.981993	+	1.70086i	−0.176371	+	0.305484i	−0.940635	−	0.339420i	\(-0.889769\pi\)
							0.764264	+	0.644904i	\(0.223103\pi\)
\(37\)	1.02771	−	0.0770165i	0.168955	−	0.0126614i	0.0100161	−	0.999950i	\(-0.496812\pi\)
							0.158939	+	0.987288i	\(0.449193\pi\)
\(41\)	−3.53981	−	4.43878i	−0.552826	−	0.693222i	0.424388	−	0.905481i	\(-0.360489\pi\)
							−0.977213	+	0.212259i	\(0.931918\pi\)
\(43\)	−3.98140	−	3.17506i	−0.607158	−	0.484193i	0.270989	−	0.962582i	\(-0.412649\pi\)
							−0.878148	+	0.478390i	\(0.841221\pi\)
\(47\)	1.88871	−	1.75247i	0.275497	−	0.255624i	−0.530308	−	0.847805i	\(-0.677924\pi\)
							0.805805	+	0.592181i	\(0.201733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.36	yes	648
8.5	even	2	inner	392.2.z.a.109.12	✓	648
49.9	even	21	inner	392.2.z.a.205.12	yes	648
392.205	even	42	inner	392.2.z.a.205.36	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.12	✓	648	8.5	even	2	inner
392.2.z.a.109.36	yes	648	1.1	even	1	trivial
392.2.z.a.205.12	yes	648	49.9	even	21	inner
392.2.z.a.205.36	yes	648	392.205	even	42	inner