Embedded newform 392.2.z.a.205.23

• Label: 392.2.z.a.205.23
• Level: $392$
• Weight: $2$
• Character: 392.205
• 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			205.23
Character	\(\chi\)	\(=\)	392.205
Dual form			392.2.z.a.109.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.569237 + 1.29459i) q^{2} +(-0.443070 + 2.93958i) q^{3} +(-1.35194 - 1.47386i) q^{4} +(3.67681 - 1.44304i) q^{5} +(-3.55334 - 2.24691i) q^{6} +(2.36353 + 1.18901i) q^{7} +(2.67762 - 0.911233i) q^{8} +(-5.57808 - 1.72061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.569237	+	1.29459i	−0.402511	+	0.915415i
							\(3\)	−0.443070	+	2.93958i	−0.255807	+	1.69717i	0.379732	+	0.925096i	\(0.376016\pi\)
−0.635539	+	0.772069i	\(0.719222\pi\)
\(5\)	3.67681	−	1.44304i	1.64432	−	0.645347i	0.650824	−	0.759229i	\(-0.274424\pi\)
							0.993494	+	0.113881i	\(0.0363283\pi\)
\(7\)	2.36353	+	1.18901i	0.893329	+	0.449404i
							\(11\)	1.30921	+	4.24434i	0.394741	+	1.27972i	0.907089	+	0.420940i	\(0.138300\pi\)
−0.512348	+	0.858778i	\(0.671224\pi\)
\(13\)	1.69877	+	0.387734i	0.471155	+	0.107538i	0.451504	−	0.892269i	\(-0.350888\pi\)
							0.0196504	+	0.999807i	\(0.493745\pi\)
\(17\)	0.673781	+	0.459376i	0.163416	+	0.111415i	0.642267	−	0.766481i	\(-0.277994\pi\)
							−0.478851	+	0.877896i	\(0.658947\pi\)
\(19\)	−5.68015	−	3.27944i	−1.30312	−	0.752354i	−0.322179	−	0.946679i	\(-0.604415\pi\)
							−0.980937	+	0.194325i	\(0.937748\pi\)
\(23\)	3.85873	−	2.63083i	0.804600	−	0.548567i	−0.0897346	−	0.995966i	\(-0.528602\pi\)
							0.894335	+	0.447399i	\(0.147650\pi\)
\(29\)	−2.32148	−	4.82061i	−0.431089	−	0.895165i	−0.997475	−	0.0710250i	\(-0.977373\pi\)
							0.566386	−	0.824140i	\(-0.308341\pi\)
\(31\)	−0.0831407	−	0.144004i	−0.0149325	−	0.0258639i	0.858463	−	0.512876i	\(-0.171420\pi\)
							−0.873395	+	0.487012i	\(0.838087\pi\)
\(37\)	−6.25284	−	0.468586i	−1.02796	−	0.0770350i	−0.449942	−	0.893058i	\(-0.648555\pi\)
							−0.578019	+	0.816023i	\(0.696174\pi\)
\(41\)	−1.52526	+	1.91261i	−0.238205	+	0.298699i	−0.886537	−	0.462659i	\(-0.846896\pi\)
							0.648332	+	0.761358i	\(0.275467\pi\)
\(43\)	−1.43122	+	1.14136i	−0.218259	+	0.174056i	−0.726516	−	0.687150i	\(-0.758862\pi\)
							0.508257	+	0.861206i	\(0.330290\pi\)
\(47\)	−5.80666	−	5.38779i	−0.846988	−	0.785890i	0.131690	−	0.991291i	\(-0.457960\pi\)
							−0.978678	+	0.205401i	\(0.934150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.205.23	yes	648
8.5	even	2	inner	392.2.z.a.205.47	yes	648
49.11	even	21	inner	392.2.z.a.109.47	yes	648
392.109	even	42	inner	392.2.z.a.109.23	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.23	✓	648	392.109	even	42	inner
392.2.z.a.109.47	yes	648	49.11	even	21	inner
392.2.z.a.205.23	yes	648	1.1	even	1	trivial
392.2.z.a.205.47	yes	648	8.5	even	2	inner