Embedded newform 171.2.g.c.121.7

• Label: 171.2.g.c.121.7
• 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			121.7
Character	\(\chi\)	\(=\)	171.121
Dual form			171.2.g.c.106.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.269545 + 0.466866i) q^{2} +(-1.40907 - 1.00723i) q^{3} +(0.854691 + 1.48037i) q^{4} -0.947325 q^{5} +(0.850050 - 0.386354i) q^{6} +(1.18430 + 2.05126i) q^{7} -1.99969 q^{8} +(0.970970 + 2.83852i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.269545	+	0.466866i	−0.190597	+	0.330124i	−0.945448	−	0.325772i	\(-0.894376\pi\)
							0.754851	+	0.655896i	\(0.227709\pi\)
\(3\)	−1.40907	−	1.00723i	−0.813528	−	0.581525i
							\(5\)	−0.947325			−0.423656			−0.211828	−	0.977307i	\(-0.567942\pi\)
−0.211828	+	0.977307i	\(0.567942\pi\)
\(7\)	1.18430	+	2.05126i	0.447622	+	0.775304i	0.998231	−	0.0594594i	\(-0.0189377\pi\)
							−0.550609	+	0.834763i	\(0.685604\pi\)
\(11\)	1.76580	+	3.05845i	0.532407	+	0.922157i	0.999284	+	0.0378342i	\(0.0120459\pi\)
							−0.466877	+	0.884322i	\(0.654621\pi\)
\(13\)	0.514122	+	0.890485i	0.142592	+	0.246976i	0.928472	−	0.371403i	\(-0.121123\pi\)
							−0.785880	+	0.618379i	\(0.787790\pi\)
\(17\)	−0.347984	−	0.602725i	−0.0843984	−	0.146182i	0.820736	−	0.571307i	\(-0.193564\pi\)
							−0.905135	+	0.425125i	\(0.860230\pi\)
\(19\)	2.46688	+	3.59367i	0.565942	+	0.824445i
							\(23\)	−1.69005	−	2.92725i	−0.352399	−	0.610374i	0.634270	−	0.773112i	\(-0.281301\pi\)
−0.986669	+	0.162738i	\(0.947967\pi\)
\(29\)	3.53104			0.655697			0.327849	−	0.944730i	\(-0.393676\pi\)
							0.327849	+	0.944730i	\(0.393676\pi\)
\(31\)	4.48279	−	7.76442i	0.805133	−	1.39453i	−0.111068	−	0.993813i	\(-0.535427\pi\)
							0.916201	−	0.400719i	\(-0.131240\pi\)
\(37\)	0.345685			0.0568303			0.0284151	−	0.999596i	\(-0.490954\pi\)
							0.0284151	+	0.999596i	\(0.490954\pi\)
\(41\)	−11.3850			−1.77803			−0.889016	−	0.457876i	\(-0.848610\pi\)
							−0.889016	+	0.457876i	\(0.848610\pi\)
\(43\)	2.10829	−	3.65166i	0.321511	−	0.556873i	−0.659289	−	0.751890i	\(-0.729143\pi\)
							0.980800	+	0.195016i	\(0.0624760\pi\)
\(47\)	10.2525			1.49548			0.747742	−	0.663990i	\(-0.231138\pi\)
							0.747742	+	0.663990i	\(0.231138\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.121.7	yes	32
3.2	odd	2		513.2.g.c.64.10		32
9.2	odd	6		513.2.h.c.235.7		32
9.7	even	3		171.2.h.c.7.10	yes	32
19.11	even	3		171.2.h.c.49.10	yes	32
57.11	odd	6		513.2.h.c.334.7		32
171.11	odd	6		513.2.g.c.505.10		32
171.106	even	3	inner	171.2.g.c.106.7	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.7	✓	32	171.106	even	3	inner
171.2.g.c.121.7	yes	32	1.1	even	1	trivial
171.2.h.c.7.10	yes	32	9.7	even	3
171.2.h.c.49.10	yes	32	19.11	even	3
513.2.g.c.64.10		32	3.2	odd	2
513.2.g.c.505.10		32	171.11	odd	6
513.2.h.c.235.7		32	9.2	odd	6
513.2.h.c.334.7		32	57.11	odd	6