Embedded newform 392.2.z.a.109.6

• Label: 392.2.z.a.109.6
• 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			109.6
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31143 - 0.529303i) q^{2} +(0.197978 + 1.31349i) q^{3} +(1.43968 + 1.38828i) q^{4} +(0.400186 + 0.157061i) q^{5} +(0.435604 - 1.82734i) q^{6} +(2.29348 - 1.31907i) q^{7} +(-1.15321 - 2.58266i) q^{8} +(1.18064 - 0.364181i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31143	−	0.529303i	−0.927318	−	0.374274i
							\(3\)	0.197978	+	1.31349i	0.114302	+	0.758347i	0.970395	+	0.241524i	\(0.0776473\pi\)
−0.856092	+	0.516823i	\(0.827115\pi\)
\(5\)	0.400186	+	0.157061i	0.178969	+	0.0702400i	0.453133	−	0.891443i	\(-0.350306\pi\)
							−0.274165	+	0.961683i	\(0.588401\pi\)
\(7\)	2.29348	−	1.31907i	0.866854	−	0.498563i
							\(11\)	1.11866	−	3.62661i	0.337289	−	1.09346i	−0.615067	−	0.788475i	\(-0.710871\pi\)
0.952356	−	0.304989i	\(-0.0986529\pi\)
\(13\)	−3.49195	+	0.797016i	−0.968494	+	0.221052i	−0.677378	−	0.735635i	\(-0.736884\pi\)
							−0.291116	+	0.956688i	\(0.594027\pi\)
\(17\)	4.88123	−	3.32797i	1.18387	−	0.807151i	0.198927	−	0.980014i	\(-0.436254\pi\)
							0.984946	+	0.172863i	\(0.0553019\pi\)
\(19\)	−1.60836	+	0.928587i	−0.368983	+	0.213033i	−0.673014	−	0.739630i	\(-0.735001\pi\)
							0.304031	+	0.952662i	\(0.401667\pi\)
\(23\)	6.85261	+	4.67203i	1.42887	+	0.974186i	0.997400	+	0.0720579i	\(0.0229566\pi\)
							0.431468	+	0.902128i	\(0.357996\pi\)
\(29\)	1.43164	−	2.97283i	0.265849	−	0.552041i	−0.724722	−	0.689041i	\(-0.758032\pi\)
							0.990571	+	0.137000i	\(0.0437462\pi\)
\(31\)	−3.59829	+	6.23242i	−0.646272	+	1.11938i	0.337735	+	0.941241i	\(0.390339\pi\)
							−0.984006	+	0.178134i	\(0.942994\pi\)
\(37\)	0.975748	−	0.0731222i	0.160412	−	0.0120212i	0.00571805	−	0.999984i	\(-0.498180\pi\)
							0.154694	+	0.987962i	\(0.450561\pi\)
\(41\)	−1.32567	−	1.66234i	−0.207035	−	0.259614i	0.667463	−	0.744643i	\(-0.267380\pi\)
							−0.874498	+	0.485029i	\(0.838809\pi\)
\(43\)	5.52298	+	4.40443i	0.842246	+	0.671669i	0.946437	−	0.322889i	\(-0.104654\pi\)
							−0.104191	+	0.994557i	\(0.533225\pi\)
\(47\)	−1.01171	+	0.938733i	−0.147574	+	0.136928i	−0.750505	−	0.660865i	\(-0.770190\pi\)
							0.602931	+	0.797793i	\(0.293999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.6	✓	648
8.5	even	2	inner	392.2.z.a.109.33	yes	648
49.9	even	21	inner	392.2.z.a.205.33	yes	648
392.205	even	42	inner	392.2.z.a.205.6	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.6	✓	648	1.1	even	1	trivial
392.2.z.a.109.33	yes	648	8.5	even	2	inner
392.2.z.a.205.6	yes	648	392.205	even	42	inner
392.2.z.a.205.33	yes	648	49.9	even	21	inner