Embedded newform 171.2.g.c.106.9

• Label: 171.2.g.c.106.9
• Level: $171$
• Weight: $2$
• Character: 171.106
• Analytic conductor: $1.365$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			106.9
Character	\(\chi\)	\(=\)	171.106
Dual form			171.2.g.c.121.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0732670 - 0.126902i) q^{2} +(0.157437 - 1.72488i) q^{3} +(0.989264 - 1.71346i) q^{4} +2.57004 q^{5} +(-0.230426 + 0.106398i) q^{6} +(-1.73898 + 3.01201i) q^{7} -0.582990 q^{8} +(-2.95043 - 0.543121i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0732670	−	0.126902i	−0.0518076	−	0.0897334i	0.838959	−	0.544195i	\(-0.183165\pi\)
							−0.890766	+	0.454462i	\(0.849832\pi\)
\(3\)	0.157437	−	1.72488i	0.0908965	−	0.995860i
							\(5\)	2.57004			1.14936			0.574678	−	0.818380i	\(-0.305127\pi\)
0.574678	+	0.818380i	\(0.305127\pi\)
\(7\)	−1.73898	+	3.01201i	−0.657274	+	1.13843i	0.324045	+	0.946042i	\(0.394957\pi\)
							−0.981319	+	0.192390i	\(0.938376\pi\)
\(11\)	2.07935	−	3.60154i	0.626947	−	1.08590i	−0.361214	−	0.932483i	\(-0.617637\pi\)
							0.988161	−	0.153421i	\(-0.0490292\pi\)
\(13\)	−2.29925	+	3.98242i	−0.637698	+	1.10452i	0.348239	+	0.937406i	\(0.386780\pi\)
							−0.985937	+	0.167119i	\(0.946554\pi\)
\(17\)	−1.50797	+	2.61188i	−0.365736	+	0.633474i	−0.988894	−	0.148622i	\(-0.952516\pi\)
							0.623158	+	0.782096i	\(0.285850\pi\)
\(19\)	3.65690	+	2.37215i	0.838951	+	0.544208i
							\(23\)	2.45941	−	4.25982i	0.512822	−	0.888234i	−0.487067	−	0.873365i	\(-0.661933\pi\)
0.999889	−	0.0148698i	\(-0.00473338\pi\)
\(29\)	−3.71403			−0.689678			−0.344839	−	0.938662i	\(-0.612066\pi\)
							−0.344839	+	0.938662i	\(0.612066\pi\)
\(31\)	3.31980	+	5.75007i	0.596254	+	1.03274i	0.993369	+	0.114973i	\(0.0366782\pi\)
							−0.397115	+	0.917769i	\(0.629988\pi\)
\(37\)	−5.28491			−0.868834			−0.434417	−	0.900712i	\(-0.643046\pi\)
							−0.434417	+	0.900712i	\(0.643046\pi\)
\(41\)	2.53193			0.395422			0.197711	−	0.980260i	\(-0.436649\pi\)
							0.197711	+	0.980260i	\(0.436649\pi\)
\(43\)	−2.39036	−	4.14022i	−0.364526	−	0.631378i	0.624174	−	0.781285i	\(-0.285436\pi\)
							−0.988700	+	0.149908i	\(0.952102\pi\)
\(47\)	9.71255			1.41672			0.708360	−	0.705851i	\(-0.249435\pi\)
							0.708360	+	0.705851i	\(0.249435\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.106.9	✓	32
3.2	odd	2		513.2.g.c.505.8		32
9.4	even	3		171.2.h.c.49.8	yes	32
9.5	odd	6		513.2.h.c.334.9		32
19.7	even	3		171.2.h.c.7.8	yes	32
57.26	odd	6		513.2.h.c.235.9		32
171.121	even	3	inner	171.2.g.c.121.9	yes	32
171.140	odd	6		513.2.g.c.64.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.9	✓	32	1.1	even	1	trivial
171.2.g.c.121.9	yes	32	171.121	even	3	inner
171.2.h.c.7.8	yes	32	19.7	even	3
171.2.h.c.49.8	yes	32	9.4	even	3
513.2.g.c.64.8		32	171.140	odd	6
513.2.g.c.505.8		32	3.2	odd	2
513.2.h.c.235.9		32	57.26	odd	6
513.2.h.c.334.9		32	9.5	odd	6