Embedded newform 207.3.f.b.160.16

• Label: 207.3.f.b.160.16
• Level: $207$
• Weight: $3$
• Character: 207.160
• 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			160.16
Character	\(\chi\)	\(=\)	207.160
Dual form			207.3.f.b.22.16

$q$-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 \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.3.f.b.22.15	✓	80	9.4	even	3	inner
207.3.f.b.22.16	yes	80	207.22	odd	6	inner
207.3.f.b.160.15	yes	80	23.22	odd	2	inner
207.3.f.b.160.16	yes	80	1.1	even	1	trivial
621.3.f.b.91.25		80	207.68	even	6
621.3.f.b.91.26		80	9.5	odd	6
621.3.f.b.505.25		80	3.2	odd	2
621.3.f.b.505.26		80	69.68	even	2