Embedded newform 171.2.g.c.106.16

• Label: 171.2.g.c.106.16
• 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			106.16
Character	\(\chi\)	\(=\)	171.106
Dual form			171.2.g.c.121.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30515 + 2.26059i) q^{2} +(1.63157 - 0.581354i) q^{3} +(-2.40684 + 4.16877i) q^{4} -2.00201 q^{5} +(3.44365 + 2.92956i) q^{6} +(0.257107 - 0.445323i) q^{7} -7.34455 q^{8} +(2.32405 - 1.89704i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30515	+	2.26059i	0.922881	+	1.59848i	0.794934	+	0.606696i	\(0.207506\pi\)
							0.127948	+	0.991781i	\(0.459161\pi\)
\(3\)	1.63157	−	0.581354i	0.941989	−	0.335645i
							\(5\)	−2.00201			−0.895325			−0.447663	−	0.894202i	\(-0.647743\pi\)
−0.447663	+	0.894202i	\(0.647743\pi\)
\(7\)	0.257107	−	0.445323i	0.0971775	−	0.168316i	−0.813338	−	0.581792i	\(-0.802352\pi\)
							0.910515	+	0.413475i	\(0.135685\pi\)
\(11\)	2.04883	−	3.54868i	0.617745	−	1.06997i	−0.372151	−	0.928172i	\(-0.621380\pi\)
							0.989896	−	0.141794i	\(-0.0452870\pi\)
\(13\)	−1.85122	+	3.20641i	−0.513437	+	0.889298i	0.486442	+	0.873713i	\(0.338294\pi\)
							−0.999879	+	0.0155853i	\(0.995039\pi\)
\(17\)	3.60664	−	6.24688i	0.874738	−	1.51509i	0.0176966	−	0.999843i	\(-0.494367\pi\)
							0.857042	−	0.515247i	\(-0.172300\pi\)
\(19\)	0.559954	+	4.32278i	0.128462	+	0.991714i
							\(23\)	−0.174335	+	0.301956i	−0.0363513	+	0.0629622i	−0.883629	−	0.468188i	\(-0.844907\pi\)
0.847277	+	0.531151i	\(0.178240\pi\)
\(29\)	−7.54782			−1.40160			−0.700798	−	0.713360i	\(-0.747172\pi\)
							−0.700798	+	0.713360i	\(0.747172\pi\)
\(31\)	−0.773246	−	1.33930i	−0.138879	−	0.240545i	0.788194	−	0.615427i	\(-0.211017\pi\)
							−0.927073	+	0.374882i	\(0.877683\pi\)
\(37\)	−6.82195			−1.12152			−0.560760	−	0.827978i	\(-0.689491\pi\)
							−0.560760	+	0.827978i	\(0.689491\pi\)
\(41\)	−2.92203			−0.456345			−0.228172	−	0.973621i	\(-0.573275\pi\)
							−0.228172	+	0.973621i	\(0.573275\pi\)
\(43\)	−0.200878	−	0.347931i	−0.0306336	−	0.0530590i	0.850302	−	0.526295i	\(-0.176419\pi\)
							−0.880936	+	0.473236i	\(0.843086\pi\)
\(47\)	6.16199			0.898819			0.449410	−	0.893326i	\(-0.351634\pi\)
							0.449410	+	0.893326i	\(0.351634\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.106.16	✓	32
3.2	odd	2		513.2.g.c.505.1		32
9.4	even	3		171.2.h.c.49.1	yes	32
9.5	odd	6		513.2.h.c.334.16		32
19.7	even	3		171.2.h.c.7.1	yes	32
57.26	odd	6		513.2.h.c.235.16		32
171.121	even	3	inner	171.2.g.c.121.16	yes	32
171.140	odd	6		513.2.g.c.64.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.16	✓	32	1.1	even	1	trivial
171.2.g.c.121.16	yes	32	171.121	even	3	inner
171.2.h.c.7.1	yes	32	19.7	even	3
171.2.h.c.49.1	yes	32	9.4	even	3
513.2.g.c.64.1		32	171.140	odd	6
513.2.g.c.505.1		32	3.2	odd	2
513.2.h.c.235.16		32	57.26	odd	6
513.2.h.c.334.16		32	9.5	odd	6