Embedded newform 171.2.g.c.121.12

• Label: 171.2.g.c.121.12
• Level: $171$
• Weight: $2$
• Character: 171.121
• 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			121.12
Character	\(\chi\)	\(=\)	171.121
Dual form			171.2.g.c.106.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.847114 - 1.46725i) q^{2} +(0.0521113 + 1.73127i) q^{3} +(-0.435206 - 0.753799i) q^{4} +0.0882176 q^{5} +(2.58434 + 1.39012i) q^{6} +(1.84695 + 3.19901i) q^{7} +1.91378 q^{8} +(-2.99457 + 0.180437i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.847114	−	1.46725i	0.599000	−	1.03750i	−0.393969	−	0.919124i	\(-0.628898\pi\)
							0.992969	−	0.118375i	\(-0.0377686\pi\)
\(3\)	0.0521113	+	1.73127i	0.0300865	+	0.999547i
							\(5\)	0.0882176			0.0394521			0.0197261	−	0.999805i	\(-0.493721\pi\)
0.0197261	+	0.999805i	\(0.493721\pi\)
\(7\)	1.84695	+	3.19901i	0.698082	+	1.20911i	0.969131	+	0.246548i	\(0.0792963\pi\)
							−0.271049	+	0.962566i	\(0.587370\pi\)
\(11\)	−1.97689	−	3.42408i	−0.596056	−	1.03240i	−0.993397	−	0.114728i	\(-0.963400\pi\)
							0.397341	−	0.917671i	\(-0.369933\pi\)
\(13\)	−2.03012	−	3.51627i	−0.563054	−	0.975238i	−0.997228	−	0.0744086i	\(-0.976293\pi\)
							0.434174	−	0.900829i	\(-0.357040\pi\)
\(17\)	−0.586674	−	1.01615i	−0.142289	−	0.246452i	0.786069	−	0.618139i	\(-0.212113\pi\)
							−0.928358	+	0.371686i	\(0.878780\pi\)
\(19\)	3.26283	+	2.89032i	0.748544	+	0.663085i
							\(23\)	−1.91604	−	3.31868i	−0.399523	−	0.691994i	0.594144	−	0.804358i	\(-0.297491\pi\)
−0.993667	+	0.112365i	\(0.964157\pi\)
\(29\)	6.56701			1.21946			0.609732	−	0.792608i	\(-0.291277\pi\)
							0.609732	+	0.792608i	\(0.291277\pi\)
\(31\)	−4.14650	+	7.18194i	−0.744733	+	1.28991i	0.205587	+	0.978639i	\(0.434090\pi\)
							−0.950320	+	0.311276i	\(0.899244\pi\)
\(37\)	−6.88356			−1.13165			−0.565825	−	0.824525i	\(-0.691442\pi\)
							−0.565825	+	0.824525i	\(0.691442\pi\)
\(41\)	−4.66112			−0.727945			−0.363972	−	0.931410i	\(-0.618580\pi\)
							−0.363972	+	0.931410i	\(0.618580\pi\)
\(43\)	4.12481	−	7.14438i	0.629028	−	1.08951i	−0.358720	−	0.933445i	\(-0.616787\pi\)
							0.987747	−	0.156062i	\(-0.0498801\pi\)
\(47\)	4.43759			0.647290			0.323645	−	0.946179i	\(-0.395092\pi\)
							0.323645	+	0.946179i	\(0.395092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.121.12	yes	32
3.2	odd	2		513.2.g.c.64.5		32
9.2	odd	6		513.2.h.c.235.12		32
9.7	even	3		171.2.h.c.7.5	yes	32
19.11	even	3		171.2.h.c.49.5	yes	32
57.11	odd	6		513.2.h.c.334.12		32
171.11	odd	6		513.2.g.c.505.5		32
171.106	even	3	inner	171.2.g.c.106.12	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.12	✓	32	171.106	even	3	inner
171.2.g.c.121.12	yes	32	1.1	even	1	trivial
171.2.h.c.7.5	yes	32	9.7	even	3
171.2.h.c.49.5	yes	32	19.11	even	3
513.2.g.c.64.5		32	3.2	odd	2
513.2.g.c.505.5		32	171.11	odd	6
513.2.h.c.235.12		32	9.2	odd	6
513.2.h.c.334.12		32	57.11	odd	6