Embedded newform 171.2.g.c.106.4

• Label: 171.2.g.c.106.4
• 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			106.4
Character	\(\chi\)	\(=\)	171.106
Dual form			171.2.g.c.121.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.888985 - 1.53977i) q^{2} +(1.15573 - 1.29008i) q^{3} +(-0.580589 + 1.00561i) q^{4} -1.27957 q^{5} +(-3.01384 - 0.632688i) q^{6} +(0.657761 - 1.13928i) q^{7} -1.49140 q^{8} +(-0.328599 - 2.98195i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.888985	−	1.53977i	−0.628607	−	1.08878i	−0.987831	−	0.155529i	\(-0.950292\pi\)
							0.359224	−	0.933251i	\(-0.383041\pi\)
\(3\)	1.15573	−	1.29008i	0.667258	−	0.744827i
							\(5\)	−1.27957			−0.572242			−0.286121	−	0.958194i	\(-0.592366\pi\)
−0.286121	+	0.958194i	\(0.592366\pi\)
\(7\)	0.657761	−	1.13928i	0.248610	−	0.430606i	−0.714530	−	0.699605i	\(-0.753359\pi\)
							0.963140	+	0.268999i	\(0.0866928\pi\)
\(11\)	−0.130095	+	0.225331i	−0.0392251	+	0.0679398i	−0.884971	−	0.465645i	\(-0.845822\pi\)
							0.845746	+	0.533585i	\(0.179156\pi\)
\(13\)	0.933961	−	1.61767i	0.259034	−	0.448660i	−0.706949	−	0.707264i	\(-0.749929\pi\)
							0.965983	+	0.258604i	\(0.0832625\pi\)
\(17\)	−0.0508308	+	0.0880415i	−0.0123283	+	0.0213532i	−0.872124	−	0.489285i	\(-0.837258\pi\)
							0.859795	+	0.510639i	\(0.170591\pi\)
\(19\)	3.11089	+	3.05326i	0.713687	+	0.700465i
							\(23\)	0.611950	−	1.05993i	0.127600	−	0.221010i	−0.795146	−	0.606418i	\(-0.792606\pi\)
0.922746	+	0.385408i	\(0.125939\pi\)
\(29\)	6.52642			1.21193			0.605963	−	0.795493i	\(-0.292788\pi\)
							0.605963	+	0.795493i	\(0.292788\pi\)
\(31\)	−0.617667	−	1.06983i	−0.110936	−	0.192147i	0.805212	−	0.592987i	\(-0.202052\pi\)
							−0.916148	+	0.400840i	\(0.868718\pi\)
\(37\)	8.59418			1.41287			0.706437	−	0.707776i	\(-0.250301\pi\)
							0.706437	+	0.707776i	\(0.250301\pi\)
\(41\)	8.21490			1.28295			0.641476	−	0.767143i	\(-0.278322\pi\)
							0.641476	+	0.767143i	\(0.278322\pi\)
\(43\)	−1.53770	−	2.66338i	−0.234498	−	0.406162i	0.724629	−	0.689139i	\(-0.242011\pi\)
							−0.959127	+	0.282977i	\(0.908678\pi\)
\(47\)	1.58093			0.230602			0.115301	−	0.993331i	\(-0.463217\pi\)
							0.115301	+	0.993331i	\(0.463217\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.106.4	✓	32
3.2	odd	2		513.2.g.c.505.13		32
9.4	even	3		171.2.h.c.49.13	yes	32
9.5	odd	6		513.2.h.c.334.4		32
19.7	even	3		171.2.h.c.7.13	yes	32
57.26	odd	6		513.2.h.c.235.4		32
171.121	even	3	inner	171.2.g.c.121.4	yes	32
171.140	odd	6		513.2.g.c.64.13		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.4	✓	32	1.1	even	1	trivial
171.2.g.c.121.4	yes	32	171.121	even	3	inner
171.2.h.c.7.13	yes	32	19.7	even	3
171.2.h.c.49.13	yes	32	9.4	even	3
513.2.g.c.64.13		32	171.140	odd	6
513.2.g.c.505.13		32	3.2	odd	2
513.2.h.c.235.4		32	57.26	odd	6
513.2.h.c.334.4		32	9.5	odd	6