Embedded newform 171.2.g.c.121.6

• Label: 171.2.g.c.121.6
• 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			121.6
Character	\(\chi\)	\(=\)	171.121
Dual form			171.2.g.c.106.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.395929 + 0.685769i) q^{2} +(0.659141 - 1.60173i) q^{3} +(0.686481 + 1.18902i) q^{4} +2.59093 q^{5} +(0.837442 + 1.08619i) q^{6} +(-0.373088 - 0.646207i) q^{7} -2.67091 q^{8} +(-2.13107 - 2.11153i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.395929	+	0.685769i	−0.279964	+	0.484912i	−0.971375	−	0.237549i	\(-0.923656\pi\)
							0.691411	+	0.722461i	\(0.256989\pi\)
\(3\)	0.659141	−	1.60173i	0.380555	−	0.924758i
							\(5\)	2.59093			1.15870			0.579349	−	0.815080i	\(-0.303307\pi\)
0.579349	+	0.815080i	\(0.303307\pi\)
\(7\)	−0.373088	−	0.646207i	−0.141014	−	0.244243i	0.786865	−	0.617126i	\(-0.211703\pi\)
							−0.927879	+	0.372882i	\(0.878370\pi\)
\(11\)	−1.28837	−	2.23153i	−0.388460	−	0.672832i	0.603783	−	0.797149i	\(-0.293659\pi\)
							−0.992243	+	0.124317i	\(0.960326\pi\)
\(13\)	3.09365	+	5.35835i	0.858023	+	1.48614i	0.873812	+	0.486265i	\(0.161641\pi\)
							−0.0157882	+	0.999875i	\(0.505026\pi\)
\(17\)	0.119999	+	0.207845i	0.0291041	+	0.0504097i	0.880211	−	0.474583i	\(-0.157401\pi\)
							−0.851107	+	0.524993i	\(0.824068\pi\)
\(19\)	3.89399	−	1.95878i	0.893343	−	0.449375i
							\(23\)	−1.93131	−	3.34513i	−0.402707	−	0.697509i	0.591345	−	0.806419i	\(-0.298597\pi\)
−0.994052	+	0.108910i	\(0.965264\pi\)
\(29\)	−6.79737			−1.26224			−0.631120	−	0.775685i	\(-0.717405\pi\)
							−0.631120	+	0.775685i	\(0.717405\pi\)
\(31\)	−3.77423	+	6.53716i	−0.677872	+	1.17411i	0.297749	+	0.954644i	\(0.403764\pi\)
							−0.975621	+	0.219464i	\(0.929569\pi\)
\(37\)	−8.47678			−1.39357			−0.696787	−	0.717278i	\(-0.745388\pi\)
							−0.696787	+	0.717278i	\(0.745388\pi\)
\(41\)	8.15194			1.27312			0.636559	−	0.771228i	\(-0.280357\pi\)
							0.636559	+	0.771228i	\(0.280357\pi\)
\(43\)	1.44011	−	2.49434i	0.219615	−	0.380384i	−0.735076	−	0.677985i	\(-0.762853\pi\)
							0.954690	+	0.297602i	\(0.0961867\pi\)
\(47\)	−4.52565			−0.660134			−0.330067	−	0.943957i	\(-0.607071\pi\)
							−0.330067	+	0.943957i	\(0.607071\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.121.6	yes	32
3.2	odd	2		513.2.g.c.64.11		32
9.2	odd	6		513.2.h.c.235.6		32
9.7	even	3		171.2.h.c.7.11	yes	32
19.11	even	3		171.2.h.c.49.11	yes	32
57.11	odd	6		513.2.h.c.334.6		32
171.11	odd	6		513.2.g.c.505.11		32
171.106	even	3	inner	171.2.g.c.106.6	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.6	✓	32	171.106	even	3	inner
171.2.g.c.121.6	yes	32	1.1	even	1	trivial
171.2.h.c.7.11	yes	32	9.7	even	3
171.2.h.c.49.11	yes	32	19.11	even	3
513.2.g.c.64.11		32	3.2	odd	2
513.2.g.c.505.11		32	171.11	odd	6
513.2.h.c.235.6		32	9.2	odd	6
513.2.h.c.334.6		32	57.11	odd	6