Embedded newform 171.3.z.a.101.33

• Label: 171.3.z.a.101.33
• Level: $171$
• Weight: $3$
• Character: 171.101
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.z (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.33
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.00730 - 0.530268i) q^{2} +(-2.74216 + 1.21678i) q^{3} +(5.00388 - 1.82126i) q^{4} +(2.67621 + 3.18938i) q^{5} +(-7.60127 + 5.11330i) q^{6} +(4.42153 + 7.65831i) q^{7} +(3.50411 - 2.02310i) q^{8} +(6.03889 - 6.67322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.00730	−	0.530268i	1.50365	−	0.265134i	0.639665	−	0.768654i	\(-0.279073\pi\)
							0.863984	+	0.503520i	\(0.167962\pi\)
\(3\)	−2.74216	+	1.21678i	−0.914053	+	0.405594i
							\(5\)	2.67621	+	3.18938i	0.535241	+	0.637876i	0.964114	−	0.265489i	\(-0.0855335\pi\)
−0.428872	+	0.903365i	\(0.641089\pi\)
\(7\)	4.42153	+	7.65831i	0.631647	+	1.09404i	0.987215	+	0.159394i	\(0.0509540\pi\)
							−0.355568	+	0.934650i	\(0.615713\pi\)
\(11\)		−	5.92985i		−	0.539077i	−0.962990	−	0.269539i	\(-0.913129\pi\)
							0.962990	−	0.269539i	\(-0.0868712\pi\)
\(13\)	4.07178	+	3.41663i	0.313214	+	0.262817i	0.785819	−	0.618457i	\(-0.212242\pi\)
							−0.472605	+	0.881274i	\(0.656686\pi\)
\(17\)	10.0372	+	11.9618i	0.590422	+	0.703638i	0.975687	−	0.219168i	\(-0.0703344\pi\)
							−0.385265	+	0.922806i	\(0.625890\pi\)
\(19\)	4.26545	−	18.5150i	0.224497	−	0.974475i
							\(23\)	−5.66327	−	15.5597i	−0.246229	−	0.676509i	−0.999816	−	0.0191611i	\(-0.993900\pi\)
0.753587	−	0.657348i	\(-0.228322\pi\)
\(29\)	−13.1982	−	36.2617i	−0.455110	−	1.25040i	−0.929085	−	0.369866i	\(-0.879403\pi\)
							0.473975	−	0.880538i	\(-0.342819\pi\)
\(31\)	−29.0735			−0.937854			−0.468927	−	0.883237i	\(-0.655359\pi\)
							−0.468927	+	0.883237i	\(0.655359\pi\)
\(37\)	23.7575			0.642094			0.321047	−	0.947063i	\(-0.395965\pi\)
							0.321047	+	0.947063i	\(0.395965\pi\)
\(41\)	−32.1956	+	5.67695i	−0.785258	+	0.138462i	−0.551879	−	0.833924i	\(-0.686089\pi\)
							−0.233378	+	0.972386i	\(0.574978\pi\)
\(43\)	−31.1609	−	11.3417i	−0.724673	−	0.263759i	−0.0467646	−	0.998906i	\(-0.514891\pi\)
							−0.677908	+	0.735147i	\(0.737113\pi\)
\(47\)	−10.4897	−	28.8202i	−0.223185	−	0.613195i	0.776676	−	0.629901i	\(-0.216904\pi\)
							−0.999860	+	0.0167055i	\(0.994682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.33	✓	228
9.5	odd	6		171.3.bf.a.158.33	yes	228
19.16	even	9		171.3.bf.a.92.33	yes	228
171.149	odd	18	inner	171.3.z.a.149.33	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.33	✓	228	1.1	even	1	trivial
171.3.z.a.149.33	yes	228	171.149	odd	18	inner
171.3.bf.a.92.33	yes	228	19.16	even	9
171.3.bf.a.158.33	yes	228	9.5	odd	6