Embedded newform 171.2.g.c.106.3

• Label: 171.2.g.c.106.3
• 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			106.3
Character	\(\chi\)	\(=\)	171.106
Dual form			171.2.g.c.121.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978515 - 1.69484i) q^{2} +(-1.73022 - 0.0795352i) q^{3} +(-0.914982 + 1.58480i) q^{4} -0.196235 q^{5} +(1.55825 + 3.01027i) q^{6} +(-2.23368 + 3.86885i) q^{7} -0.332766 q^{8} +(2.98735 + 0.275227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.978515	−	1.69484i	−0.691914	−	1.19843i	−0.971210	−	0.238226i	\(-0.923434\pi\)
							0.279295	−	0.960205i	\(-0.409899\pi\)
\(3\)	−1.73022	−	0.0795352i	−0.998945	−	0.0459197i
							\(5\)	−0.196235			−0.0877587			−0.0438794	−	0.999037i	\(-0.513972\pi\)
−0.0438794	+	0.999037i	\(0.513972\pi\)
\(7\)	−2.23368	+	3.86885i	−0.844253	+	1.46229i	0.0420154	+	0.999117i	\(0.486622\pi\)
							−0.886268	+	0.463172i	\(0.846711\pi\)
\(11\)	0.0755474	−	0.130852i	0.0227784	−	0.0394534i	−0.854412	−	0.519597i	\(-0.826082\pi\)
							0.877190	+	0.480144i	\(0.159415\pi\)
\(13\)	−0.234555	+	0.406262i	−0.0650539	+	0.112677i	−0.896718	−	0.442603i	\(-0.854055\pi\)
							0.831664	+	0.555279i	\(0.187389\pi\)
\(17\)	0.441434	−	0.764587i	0.107064	−	0.185440i	−0.807516	−	0.589846i	\(-0.799188\pi\)
							0.914579	+	0.404406i	\(0.132522\pi\)
\(19\)	−0.132216	+	4.35689i	−0.0303325	+	0.999540i
							\(23\)	−2.57631	+	4.46230i	−0.537197	+	0.930453i	0.461856	+	0.886955i	\(0.347184\pi\)
−0.999054	+	0.0434982i	\(0.986150\pi\)
\(29\)	−1.57810			−0.293047			−0.146523	−	0.989207i	\(-0.546808\pi\)
							−0.146523	+	0.989207i	\(0.546808\pi\)
\(31\)	−1.37848	−	2.38759i	−0.247582	−	0.428824i	0.715273	−	0.698845i	\(-0.246302\pi\)
							−0.962854	+	0.270022i	\(0.912969\pi\)
\(37\)	−1.36950			−0.225145			−0.112572	−	0.993644i	\(-0.535909\pi\)
							−0.112572	+	0.993644i	\(0.535909\pi\)
\(41\)	−8.13338			−1.27022			−0.635110	−	0.772422i	\(-0.719045\pi\)
							−0.635110	+	0.772422i	\(0.719045\pi\)
\(43\)	4.09593	+	7.09436i	0.624623	+	1.08188i	0.988614	+	0.150477i	\(0.0480808\pi\)
							−0.363990	+	0.931403i	\(0.618586\pi\)
\(47\)	−11.8802			−1.73291			−0.866453	−	0.499259i	\(-0.833606\pi\)
							−0.866453	+	0.499259i	\(0.833606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.106.3	✓	32
3.2	odd	2		513.2.g.c.505.14		32
9.4	even	3		171.2.h.c.49.14	yes	32
9.5	odd	6		513.2.h.c.334.3		32
19.7	even	3		171.2.h.c.7.14	yes	32
57.26	odd	6		513.2.h.c.235.3		32
171.121	even	3	inner	171.2.g.c.121.3	yes	32
171.140	odd	6		513.2.g.c.64.14		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.3	✓	32	1.1	even	1	trivial
171.2.g.c.121.3	yes	32	171.121	even	3	inner
171.2.h.c.7.14	yes	32	19.7	even	3
171.2.h.c.49.14	yes	32	9.4	even	3
513.2.g.c.64.14		32	171.140	odd	6
513.2.g.c.505.14		32	3.2	odd	2
513.2.h.c.235.3		32	57.26	odd	6
513.2.h.c.334.3		32	9.5	odd	6