Embedded newform 171.2.g.c.106.14

• Label: 171.2.g.c.106.14
• 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			106.14
Character	\(\chi\)	\(=\)	171.106
Dual form			171.2.g.c.121.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19971 + 2.07797i) q^{2} +(-0.340918 + 1.69817i) q^{3} +(-1.87863 + 3.25388i) q^{4} +0.719678 q^{5} +(-3.93774 + 1.32890i) q^{6} +(1.65862 - 2.87282i) q^{7} -4.21641 q^{8} +(-2.76755 - 1.15787i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19971	+	2.07797i	0.848326	+	1.46934i	0.882701	+	0.469935i	\(0.155723\pi\)
							−0.0343750	+	0.999409i	\(0.510944\pi\)
\(3\)	−0.340918	+	1.69817i	−0.196829	+	0.980438i
							\(5\)	0.719678			0.321850			0.160925	−	0.986967i	\(-0.448552\pi\)
0.160925	+	0.986967i	\(0.448552\pi\)
\(7\)	1.65862	−	2.87282i	0.626900	−	1.08582i	−0.361270	−	0.932461i	\(-0.617657\pi\)
							0.988170	−	0.153362i	\(-0.0490101\pi\)
\(11\)	0.550465	−	0.953433i	0.165971	−	0.287471i	−0.771028	−	0.636801i	\(-0.780257\pi\)
							0.937000	+	0.349330i	\(0.113591\pi\)
\(13\)	2.37472	−	4.11314i	0.658630	−	1.14078i	−0.322341	−	0.946624i	\(-0.604470\pi\)
							0.980971	−	0.194157i	\(-0.0621970\pi\)
\(17\)	−3.13488	+	5.42977i	−0.760320	+	1.31691i	0.182366	+	0.983231i	\(0.441624\pi\)
							−0.942686	+	0.333682i	\(0.891709\pi\)
\(19\)	−4.19132	+	1.19702i	−0.961554	+	0.274615i
							\(23\)	1.11407	−	1.92963i	0.232300	−	0.402355i	−0.726185	−	0.687500i	\(-0.758708\pi\)
0.958485	+	0.285145i	\(0.0920416\pi\)
\(29\)	5.94195			1.10339			0.551696	−	0.834045i	\(-0.313981\pi\)
							0.551696	+	0.834045i	\(0.313981\pi\)
\(31\)	−0.763210	−	1.32192i	−0.137077	−	0.237424i	0.789312	−	0.613992i	\(-0.210437\pi\)
							−0.926389	+	0.376568i	\(0.877104\pi\)
\(37\)	−3.69507			−0.607465			−0.303733	−	0.952757i	\(-0.598233\pi\)
							−0.303733	+	0.952757i	\(0.598233\pi\)
\(41\)	5.68106			0.887232			0.443616	−	0.896217i	\(-0.353695\pi\)
							0.443616	+	0.896217i	\(0.353695\pi\)
\(43\)	−2.30746	−	3.99663i	−0.351884	−	0.609480i	0.634696	−	0.772762i	\(-0.281125\pi\)
							−0.986579	+	0.163282i	\(0.947792\pi\)
\(47\)	0.283238			0.0413144			0.0206572	−	0.999787i	\(-0.493424\pi\)
							0.0206572	+	0.999787i	\(0.493424\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.106.14	✓	32
3.2	odd	2		513.2.g.c.505.3		32
9.4	even	3		171.2.h.c.49.3	yes	32
9.5	odd	6		513.2.h.c.334.14		32
19.7	even	3		171.2.h.c.7.3	yes	32
57.26	odd	6		513.2.h.c.235.14		32
171.121	even	3	inner	171.2.g.c.121.14	yes	32
171.140	odd	6		513.2.g.c.64.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.14	✓	32	1.1	even	1	trivial
171.2.g.c.121.14	yes	32	171.121	even	3	inner
171.2.h.c.7.3	yes	32	19.7	even	3
171.2.h.c.49.3	yes	32	9.4	even	3
513.2.g.c.64.3		32	171.140	odd	6
513.2.g.c.505.3		32	3.2	odd	2
513.2.h.c.235.14		32	57.26	odd	6
513.2.h.c.334.14		32	9.5	odd	6