Embedded newform 171.2.g.c.106.15

• Label: 171.2.g.c.106.15
• 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			106.15
Character	\(\chi\)	\(=\)	171.106
Dual form			171.2.g.c.121.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30160 + 2.25443i) q^{2} +(-1.17382 - 1.27363i) q^{3} +(-2.38830 + 4.13666i) q^{4} +2.87794 q^{5} +(1.34348 - 4.30405i) q^{6} +(-1.80240 + 3.12185i) q^{7} -7.22804 q^{8} +(-0.244287 + 2.99004i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30160	+	2.25443i	0.920367	+	1.59412i	0.798847	+	0.601534i	\(0.205443\pi\)
							0.121520	+	0.992589i	\(0.461223\pi\)
\(3\)	−1.17382	−	1.27363i	−0.677706	−	0.735333i
							\(5\)	2.87794			1.28706			0.643528	−	0.765423i	\(-0.277470\pi\)
0.643528	+	0.765423i	\(0.277470\pi\)
\(7\)	−1.80240	+	3.12185i	−0.681243	+	1.17995i	0.293359	+	0.956002i	\(0.405227\pi\)
							−0.974602	+	0.223945i	\(0.928106\pi\)
\(11\)	−0.154676	+	0.267906i	−0.0466364	+	0.0807767i	−0.888401	−	0.459068i	\(-0.848184\pi\)
							0.841765	+	0.539844i	\(0.181517\pi\)
\(13\)	2.73977	−	4.74542i	0.759875	−	1.31614i	−0.183039	−	0.983106i	\(-0.558593\pi\)
							0.942914	−	0.333036i	\(-0.108073\pi\)
\(17\)	1.60627	−	2.78214i	0.389578	−	0.674769i	−0.602815	−	0.797881i	\(-0.705954\pi\)
							0.992393	+	0.123112i	\(0.0392875\pi\)
\(19\)	2.06434	−	3.83907i	0.473593	−	0.880744i
							\(23\)	−0.598787	+	1.03713i	−0.124856	+	0.216256i	−0.921677	−	0.387959i	\(-0.873180\pi\)
0.796821	+	0.604216i	\(0.206513\pi\)
\(29\)	−3.08188			−0.572290			−0.286145	−	0.958186i	\(-0.592374\pi\)
							−0.286145	+	0.958186i	\(0.592374\pi\)
\(31\)	−0.960973	−	1.66445i	−0.172596	−	0.298945i	0.766731	−	0.641969i	\(-0.221882\pi\)
							−0.939327	+	0.343024i	\(0.888549\pi\)
\(37\)	8.85095			1.45509			0.727544	−	0.686061i	\(-0.240662\pi\)
							0.727544	+	0.686061i	\(0.240662\pi\)
\(41\)	−8.37681			−1.30824			−0.654119	−	0.756392i	\(-0.726960\pi\)
							−0.654119	+	0.756392i	\(0.726960\pi\)
\(43\)	0.880584	+	1.52522i	0.134288	+	0.232593i	0.925325	−	0.379175i	\(-0.123792\pi\)
							−0.791037	+	0.611768i	\(0.790459\pi\)
\(47\)	−0.974401			−0.142131			−0.0710655	−	0.997472i	\(-0.522640\pi\)
							−0.0710655	+	0.997472i	\(0.522640\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.106.15	✓	32
3.2	odd	2		513.2.g.c.505.2		32
9.4	even	3		171.2.h.c.49.2	yes	32
9.5	odd	6		513.2.h.c.334.15		32
19.7	even	3		171.2.h.c.7.2	yes	32
57.26	odd	6		513.2.h.c.235.15		32
171.121	even	3	inner	171.2.g.c.121.15	yes	32
171.140	odd	6		513.2.g.c.64.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.15	✓	32	1.1	even	1	trivial
171.2.g.c.121.15	yes	32	171.121	even	3	inner
171.2.h.c.7.2	yes	32	19.7	even	3
171.2.h.c.49.2	yes	32	9.4	even	3
513.2.g.c.64.2		32	171.140	odd	6
513.2.g.c.505.2		32	3.2	odd	2
513.2.h.c.235.15		32	57.26	odd	6
513.2.h.c.334.15		32	9.5	odd	6