Embedded newform 171.3.i.a.103.12

• Label: 171.3.i.a.103.12
• Level: $171$
• Weight: $3$
• Character: 171.103
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			103.12
Character	\(\chi\)	\(=\)	171.103
Dual form			171.3.i.a.88.27

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.77221i q^{2} +(1.70070 + 2.47136i) q^{3} +0.859268 q^{4} +(-0.983566 - 1.70359i) q^{5} +(4.37977 - 3.01400i) q^{6} +(6.71110 + 11.6240i) q^{7} -8.61165i q^{8} +(-3.21525 + 8.40608i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 3 q^{3} - 146 q^{4} + q^{5} + 7 q^{6} - 3 q^{7} - 13 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.77221i		−	0.886105i	−0.896496	−	0.443053i	\(-0.853895\pi\)
							0.896496	−	0.443053i	\(-0.146105\pi\)
\(3\)	1.70070	+	2.47136i	0.566900	+	0.823787i
							\(5\)	−0.983566	−	1.70359i	−0.196713	−	0.340717i	0.750748	−	0.660589i	\(-0.229693\pi\)
−0.947461	+	0.319872i	\(0.896360\pi\)
\(7\)	6.71110	+	11.6240i	0.958728	+	1.66057i	0.725596	+	0.688121i	\(0.241564\pi\)
							0.233132	+	0.972445i	\(0.425103\pi\)
\(11\)	−0.278683	−	0.482693i	−0.0253348	−	0.0438812i	0.853080	−	0.521780i	\(-0.174732\pi\)
							−0.878415	+	0.477899i	\(0.841399\pi\)
\(13\)		−	1.13783i		−	0.0875254i	−0.999042	−	0.0437627i	\(-0.986065\pi\)
							0.999042	−	0.0437627i	\(-0.0139346\pi\)
\(17\)	−3.70565	+	6.41838i	−0.217980	+	0.377552i	−0.954190	−	0.299201i	\(-0.903280\pi\)
							0.736211	+	0.676752i	\(0.236613\pi\)
\(19\)	16.6559	−	9.14222i	0.876628	−	0.481169i
							\(23\)	23.2477			1.01077			0.505385	−	0.862894i	\(-0.331351\pi\)
0.505385	+	0.862894i	\(0.331351\pi\)
\(29\)	−24.5109	−	14.1514i	−0.845204	−	0.487979i	0.0138255	−	0.999904i	\(-0.495599\pi\)
							−0.859030	+	0.511925i	\(0.828932\pi\)
\(31\)	−36.5694	−	21.1134i	−1.17966	−	0.681076i	−0.223723	−	0.974653i	\(-0.571821\pi\)
							−0.955935	+	0.293577i	\(0.905154\pi\)
\(37\)			7.57484i			0.204725i	0.994747	+	0.102363i	\(0.0326402\pi\)
							−0.994747	+	0.102363i	\(0.967360\pi\)
\(41\)	11.0682	−	6.39025i	0.269957	−	0.155860i	−0.358911	−	0.933372i	\(-0.616852\pi\)
							0.628868	+	0.777512i	\(0.283519\pi\)
\(43\)	−13.1067			−0.304808			−0.152404	−	0.988318i	\(-0.548701\pi\)
							−0.152404	+	0.988318i	\(0.548701\pi\)
\(47\)	31.7451	−	54.9841i	0.675427	−	1.16987i	−0.300916	−	0.953651i	\(-0.597292\pi\)
							0.976344	−	0.216224i	\(-0.0693742\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.i.a.103.12	yes	76
3.2	odd	2		513.3.i.a.388.27		76
9.2	odd	6		513.3.s.a.46.27		76
9.7	even	3		171.3.s.a.160.12	yes	76
19.12	odd	6		171.3.s.a.31.12	yes	76
57.50	even	6		513.3.s.a.145.27		76
171.88	odd	6	inner	171.3.i.a.88.27	✓	76
171.164	even	6		513.3.i.a.316.12		76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.i.a.88.27	✓	76	171.88	odd	6	inner
171.3.i.a.103.12	yes	76	1.1	even	1	trivial
171.3.s.a.31.12	yes	76	19.12	odd	6
171.3.s.a.160.12	yes	76	9.7	even	3
513.3.i.a.316.12		76	171.164	even	6
513.3.i.a.388.27		76	3.2	odd	2
513.3.s.a.46.27		76	9.2	odd	6
513.3.s.a.145.27		76	57.50	even	6