Embedded newform 207.3.f.b.22.16

• Label: 207.3.f.b.22.16
• Level: $207$
• Weight: $3$
• Character: 207.22
• Analytic conductor: $5.640$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	207.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.64034147226\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			22.16
Character	\(\chi\)	\(=\)	207.22
Dual form			207.3.f.b.160.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.740154 - 1.28198i) q^{2} +(0.0693864 - 2.99920i) q^{3} +(0.904345 - 1.56637i) q^{4} +(2.25399 + 1.30134i) q^{5} +(-3.89628 + 2.13091i) q^{6} +(-10.4541 + 6.03568i) q^{7} -8.59865 q^{8} +(-8.99037 - 0.416207i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 2 q^{2} - 6 q^{3} - 66 q^{4} - 42 q^{6} - 16 q^{8} - 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.740154	−	1.28198i	−0.370077	−	0.640992i	0.619500	−	0.784997i	\(-0.287335\pi\)
							−0.989577	+	0.144005i	\(0.954002\pi\)
\(3\)	0.0693864	−	2.99920i	0.0231288	−	0.999732i
							\(5\)	2.25399	+	1.30134i	0.450798	+	0.260268i	0.708167	−	0.706045i	\(-0.249522\pi\)
−0.257369	+	0.966313i	\(0.582856\pi\)
\(7\)	−10.4541	+	6.03568i	−1.49344	+	0.862240i	−0.999972	−	0.00752158i	\(-0.997606\pi\)
							−0.493472	+	0.869762i	\(0.664272\pi\)
\(11\)	−7.58482	+	4.37910i	−0.689529	+	0.398100i	−0.803436	−	0.595392i	\(-0.796997\pi\)
							0.113907	+	0.993491i	\(0.463664\pi\)
\(13\)	3.26924	−	5.66250i	0.251480	−	0.435577i	−0.712453	−	0.701720i	\(-0.752416\pi\)
							0.963934	+	0.266143i	\(0.0857493\pi\)
\(17\)		−	6.61662i		−	0.389213i	−0.980881	−	0.194607i	\(-0.937657\pi\)
							0.980881	−	0.194607i	\(-0.0623430\pi\)
\(19\)		−	21.9764i		−	1.15665i	−0.815806	−	0.578326i	\(-0.803706\pi\)
							0.815806	−	0.578326i	\(-0.196294\pi\)
\(23\)	−18.8496	+	13.1793i	−0.819546	+	0.573014i
							\(29\)	3.97975	+	6.89313i	0.137233	+	0.237694i	0.926448	−	0.376422i	\(-0.122846\pi\)
−0.789215	+	0.614117i	\(0.789512\pi\)
\(31\)	13.7260	−	23.7741i	0.442773	−	0.766906i	−0.555121	−	0.831770i	\(-0.687328\pi\)
							0.997894	+	0.0648640i	\(0.0206614\pi\)
\(37\)		−	33.5379i		−	0.906431i	−0.891401	−	0.453215i	\(-0.850277\pi\)
							0.891401	−	0.453215i	\(-0.149723\pi\)
\(41\)	11.8357	−	20.5001i	0.288677	−	0.500002i	−0.684818	−	0.728715i	\(-0.740118\pi\)
							0.973494	+	0.228712i	\(0.0734515\pi\)
\(43\)	22.6979	−	13.1047i	0.527859	−	0.304759i	−0.212285	−	0.977208i	\(-0.568091\pi\)
							0.740144	+	0.672448i	\(0.234757\pi\)
\(47\)	0.900001	+	1.55885i	0.0191489	+	0.0331670i	0.875441	−	0.483325i	\(-0.160571\pi\)
							−0.856292	+	0.516492i	\(0.827238\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.3.f.b.22.16	yes	80
3.2	odd	2		621.3.f.b.91.25		80
9.2	odd	6		621.3.f.b.505.26		80
9.7	even	3	inner	207.3.f.b.160.15	yes	80
23.22	odd	2	inner	207.3.f.b.22.15	✓	80
69.68	even	2		621.3.f.b.91.26		80
207.137	even	6		621.3.f.b.505.25		80
207.160	odd	6	inner	207.3.f.b.160.16	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.3.f.b.22.15	✓	80	23.22	odd	2	inner
207.3.f.b.22.16	yes	80	1.1	even	1	trivial
207.3.f.b.160.15	yes	80	9.7	even	3	inner
207.3.f.b.160.16	yes	80	207.160	odd	6	inner
621.3.f.b.91.25		80	3.2	odd	2
621.3.f.b.91.26		80	69.68	even	2
621.3.f.b.505.25		80	207.137	even	6
621.3.f.b.505.26		80	9.2	odd	6