Embedded newform 171.2.g.c.121.13

• Label: 171.2.g.c.121.13
• 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			121.13
Character	\(\chi\)	\(=\)	171.121
Dual form			171.2.g.c.106.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.01016 - 1.74964i) q^{2} +(-1.70377 - 0.311691i) q^{3} +(-1.04083 - 1.80277i) q^{4} -4.18739 q^{5} +(-2.26643 + 2.66614i) q^{6} +(-0.976107 - 1.69067i) q^{7} -0.164982 q^{8} +(2.80570 + 1.06210i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.01016	−	1.74964i	0.714288	−	1.23718i	−0.248945	−	0.968518i	\(-0.580084\pi\)
							0.963233	−	0.268666i	\(-0.0865828\pi\)
\(3\)	−1.70377	−	0.311691i	−0.983675	−	0.179955i
							\(5\)	−4.18739			−1.87266			−0.936328	−	0.351127i	\(-0.885799\pi\)
−0.936328	+	0.351127i	\(0.885799\pi\)
\(7\)	−0.976107	−	1.69067i	−0.368934	−	0.639012i	0.620465	−	0.784234i	\(-0.286944\pi\)
							−0.989399	+	0.145222i	\(0.953610\pi\)
\(11\)	−0.669450	−	1.15952i	−0.201847	−	0.349609i	0.747277	−	0.664513i	\(-0.231361\pi\)
							−0.949124	+	0.314904i	\(0.898028\pi\)
\(13\)	−0.975550	−	1.68970i	−0.270569	−	0.468639i	0.698439	−	0.715670i	\(-0.253878\pi\)
							−0.969008	+	0.247031i	\(0.920545\pi\)
\(17\)	−3.34047	−	5.78587i	−0.810184	−	1.40328i	−0.912735	−	0.408551i	\(-0.866034\pi\)
							0.102552	−	0.994728i	\(-0.467299\pi\)
\(19\)	4.11317	+	1.44285i	0.943626	+	0.331012i
							\(23\)	0.986266	+	1.70826i	0.205651	+	0.356197i	0.950340	−	0.311214i	\(-0.100736\pi\)
−0.744689	+	0.667411i	\(0.767402\pi\)
\(29\)	−5.90062			−1.09572			−0.547859	−	0.836571i	\(-0.684557\pi\)
							−0.547859	+	0.836571i	\(0.684557\pi\)
\(31\)	−0.385886	+	0.668373i	−0.0693071	+	0.120043i	−0.898596	−	0.438776i	\(-0.855412\pi\)
							0.829289	+	0.558819i	\(0.188746\pi\)
\(37\)	2.26011			0.371560			0.185780	−	0.982591i	\(-0.440519\pi\)
							0.185780	+	0.982591i	\(0.440519\pi\)
\(41\)	7.59691			1.18644			0.593219	−	0.805041i	\(-0.297857\pi\)
							0.593219	+	0.805041i	\(0.297857\pi\)
\(43\)	−3.97802	+	6.89012i	−0.606642	+	1.05073i	0.385148	+	0.922855i	\(0.374150\pi\)
							−0.991790	+	0.127879i	\(0.959183\pi\)
\(47\)	−1.10774			−0.161580			−0.0807902	−	0.996731i	\(-0.525744\pi\)
							−0.0807902	+	0.996731i	\(0.525744\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.121.13	yes	32
3.2	odd	2		513.2.g.c.64.4		32
9.2	odd	6		513.2.h.c.235.13		32
9.7	even	3		171.2.h.c.7.4	yes	32
19.11	even	3		171.2.h.c.49.4	yes	32
57.11	odd	6		513.2.h.c.334.13		32
171.11	odd	6		513.2.g.c.505.4		32
171.106	even	3	inner	171.2.g.c.106.13	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.13	✓	32	171.106	even	3	inner
171.2.g.c.121.13	yes	32	1.1	even	1	trivial
171.2.h.c.7.4	yes	32	9.7	even	3
171.2.h.c.49.4	yes	32	19.11	even	3
513.2.g.c.64.4		32	3.2	odd	2
513.2.g.c.505.4		32	171.11	odd	6
513.2.h.c.235.13		32	9.2	odd	6
513.2.h.c.334.13		32	57.11	odd	6