Embedded newform 392.2.z.a.109.3

• Label: 392.2.z.a.109.3
• 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			109.3
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41164 - 0.0853275i) q^{2} +(-0.290045 - 1.92432i) q^{3} +(1.98544 + 0.240903i) q^{4} +(-1.10812 - 0.434904i) q^{5} +(0.245241 + 2.74119i) q^{6} +(1.77530 + 1.96171i) q^{7} +(-2.78216 - 0.509480i) q^{8} +(-0.752171 + 0.232014i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41164	−	0.0853275i	−0.998178	−	0.0603356i
							\(3\)	−0.290045	−	1.92432i	−0.167457	−	1.11101i	−0.901776	−	0.432203i	\(-0.857736\pi\)
0.734319	−	0.678805i	\(-0.237502\pi\)
\(5\)	−1.10812	−	0.434904i	−0.495565	−	0.194495i	0.104378	−	0.994538i	\(-0.466715\pi\)
							−0.599943	+	0.800043i	\(0.704810\pi\)
\(7\)	1.77530	+	1.96171i	0.671002	+	0.741456i
							\(11\)	1.15441	−	3.74250i	0.348067	−	1.12841i	−0.597195	−	0.802096i	\(-0.703718\pi\)
0.945263	−	0.326310i	\(-0.105805\pi\)
\(13\)	4.65240	−	1.06188i	1.29034	−	0.294513i	0.478345	−	0.878172i	\(-0.341237\pi\)
							0.811999	+	0.583659i	\(0.198380\pi\)
\(17\)	2.11461	−	1.44172i	0.512868	−	0.349668i	−0.279068	−	0.960271i	\(-0.590026\pi\)
							0.791936	+	0.610604i	\(0.209073\pi\)
\(19\)	−3.04750	+	1.75947i	−0.699144	+	0.403651i	−0.807028	−	0.590513i	\(-0.798926\pi\)
							0.107885	+	0.994163i	\(0.465592\pi\)
\(23\)	−3.64260	−	2.48348i	−0.759534	−	0.517842i	0.120522	−	0.992711i	\(-0.461543\pi\)
							−0.880057	+	0.474869i	\(0.842495\pi\)
\(29\)	−2.00797	+	4.16959i	−0.372871	+	0.774274i	−0.999989	−	0.00465011i	\(-0.998520\pi\)
							0.627119	+	0.778924i	\(0.284234\pi\)
\(31\)	1.49610	−	2.59132i	0.268708	−	0.465416i	−0.699821	−	0.714319i	\(-0.746737\pi\)
							0.968528	+	0.248903i	\(0.0800700\pi\)
\(37\)	1.67124	−	0.125242i	0.274750	−	0.0205896i	0.0633572	−	0.997991i	\(-0.479819\pi\)
							0.211392	+	0.977401i	\(0.432200\pi\)
\(41\)	1.46431	+	1.83619i	0.228687	+	0.286764i	0.882915	−	0.469533i	\(-0.155578\pi\)
							−0.654228	+	0.756297i	\(0.727006\pi\)
\(43\)	−6.23910	−	4.97552i	−0.951454	−	0.758760i	0.0190561	−	0.999818i	\(-0.493934\pi\)
							−0.970510	+	0.241059i	\(0.922505\pi\)
\(47\)	8.15459	−	7.56636i	1.18947	−	1.10367i	0.197098	−	0.980384i	\(-0.436848\pi\)
							0.992372	−	0.123283i	\(-0.0393421\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.3	✓	648
8.5	even	2	inner	392.2.z.a.109.28	yes	648
49.9	even	21	inner	392.2.z.a.205.28	yes	648
392.205	even	42	inner	392.2.z.a.205.3	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.3	✓	648	1.1	even	1	trivial
392.2.z.a.109.28	yes	648	8.5	even	2	inner
392.2.z.a.205.3	yes	648	392.205	even	42	inner
392.2.z.a.205.28	yes	648	49.9	even	21	inner