Embedded newform 171.2.g.c.121.10

• Label: 171.2.g.c.121.10
• 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			121.10
Character	\(\chi\)	\(=\)	171.121
Dual form			171.2.g.c.106.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0973467 - 0.168609i) q^{2} +(1.55267 - 0.767609i) q^{3} +(0.981047 + 1.69922i) q^{4} -1.90563 q^{5} +(0.0217210 - 0.336519i) q^{6} +(1.69446 + 2.93489i) q^{7} +0.771394 q^{8} +(1.82155 - 2.38368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0973467	−	0.168609i	0.0688345	−	0.119225i	−0.829554	−	0.558426i	\(-0.811405\pi\)
							0.898389	+	0.439202i	\(0.144739\pi\)
\(3\)	1.55267	−	0.767609i	0.896433	−	0.443179i
							\(5\)	−1.90563			−0.852225			−0.426113	−	0.904670i	\(-0.640117\pi\)
−0.426113	+	0.904670i	\(0.640117\pi\)
\(7\)	1.69446	+	2.93489i	0.640445	+	1.10928i	0.985334	+	0.170639i	\(0.0545833\pi\)
							−0.344889	+	0.938644i	\(0.612083\pi\)
\(11\)	−0.311589	−	0.539688i	−0.0939477	−	0.162722i	0.815221	−	0.579150i	\(-0.196615\pi\)
							−0.909169	+	0.416428i	\(0.863282\pi\)
\(13\)	−1.84489	−	3.19545i	−0.511681	−	0.886258i	−0.999908	−	0.0135411i	\(-0.995690\pi\)
							0.488227	−	0.872717i	\(-0.337644\pi\)
\(17\)	−3.04830	−	5.27981i	−0.739321	−	1.28054i	−0.952802	−	0.303594i	\(-0.901813\pi\)
							0.213481	−	0.976947i	\(-0.431520\pi\)
\(19\)	−1.14464	−	4.20592i	−0.262599	−	0.964905i
							\(23\)	3.92442	+	6.79729i	0.818297	+	1.41733i	0.906936	+	0.421269i	\(0.138415\pi\)
−0.0886387	+	0.996064i	\(0.528252\pi\)
\(29\)	−1.18459			−0.219972			−0.109986	−	0.993933i	\(-0.535081\pi\)
							−0.109986	+	0.993933i	\(0.535081\pi\)
\(31\)	−0.910124	+	1.57638i	−0.163463	+	0.283126i	−0.936108	−	0.351711i	\(-0.885600\pi\)
							0.772645	+	0.634838i	\(0.218933\pi\)
\(37\)	5.63896			0.927039			0.463520	−	0.886087i	\(-0.346586\pi\)
							0.463520	+	0.886087i	\(0.346586\pi\)
\(41\)	−4.03639			−0.630378			−0.315189	−	0.949029i	\(-0.602068\pi\)
							−0.315189	+	0.949029i	\(0.602068\pi\)
\(43\)	−2.54719	+	4.41186i	−0.388442	+	0.672802i	−0.992240	−	0.124335i	\(-0.960320\pi\)
							0.603798	+	0.797138i	\(0.293653\pi\)
\(47\)	−12.8718			−1.87754			−0.938772	−	0.344539i	\(-0.888035\pi\)
							−0.938772	+	0.344539i	\(0.888035\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.121.10	yes	32
3.2	odd	2		513.2.g.c.64.7		32
9.2	odd	6		513.2.h.c.235.10		32
9.7	even	3		171.2.h.c.7.7	yes	32
19.11	even	3		171.2.h.c.49.7	yes	32
57.11	odd	6		513.2.h.c.334.10		32
171.11	odd	6		513.2.g.c.505.7		32
171.106	even	3	inner	171.2.g.c.106.10	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.10	✓	32	171.106	even	3	inner
171.2.g.c.121.10	yes	32	1.1	even	1	trivial
171.2.h.c.7.7	yes	32	9.7	even	3
171.2.h.c.49.7	yes	32	19.11	even	3
513.2.g.c.64.7		32	3.2	odd	2
513.2.g.c.505.7		32	171.11	odd	6
513.2.h.c.235.10		32	9.2	odd	6
513.2.h.c.334.10		32	57.11	odd	6