Embedded newform 171.4.e.a.58.19

• Label: 171.4.e.a.58.19
• Level: $171$
• Weight: $4$
• Character: 171.58
• Analytic conductor: $10.089$
• Analytic rank: $0$
• Dimension: $54$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0893266110\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			58.19
Character	\(\chi\)	\(=\)	171.58
Dual form			171.4.e.a.115.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.989966 + 1.71467i) q^{2} +(4.88583 + 1.76881i) q^{3} +(2.03993 - 3.53327i) q^{4} +(1.25650 - 2.17632i) q^{5} +(1.80387 + 10.1287i) q^{6} +(-18.0950 - 31.3414i) q^{7} +23.9173 q^{8} +(20.7426 + 17.2842i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} + 2 q^{3} - 108 q^{4} - 26 q^{5} + 59 q^{6} + 138 q^{8} - 102 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.989966	+	1.71467i	0.350006	+	0.606228i	0.986250	−	0.165260i	\(-0.0528463\pi\)
							−0.636244	+	0.771488i	\(0.719513\pi\)
\(3\)	4.88583	+	1.76881i	0.940278	+	0.340408i
							\(5\)	1.25650	−	2.17632i	0.112385	−	0.194656i	−0.804347	−	0.594160i	\(-0.797484\pi\)
0.916731	+	0.399504i	\(0.130818\pi\)
\(7\)	−18.0950	−	31.3414i	−0.977036	−	1.69228i	−0.673044	−	0.739602i	\(-0.735014\pi\)
							−0.303992	−	0.952674i	\(-0.598320\pi\)
\(11\)	17.8332	+	30.8880i	0.488810	+	0.846645i	0.999917	−	0.0128727i	\(-0.00409763\pi\)
							−0.511107	+	0.859517i	\(0.670764\pi\)
\(13\)	32.9774	−	57.1185i	0.703560	−	1.21860i	−0.263649	−	0.964619i	\(-0.584926\pi\)
							0.967209	−	0.253983i	\(-0.0817407\pi\)
\(17\)	27.0750			0.386274			0.193137	−	0.981172i	\(-0.438134\pi\)
							0.193137	+	0.981172i	\(0.438134\pi\)
\(19\)	−19.0000			−0.229416
							\(23\)	−78.7214	+	136.349i	−0.713676	+	1.23612i	0.249792	+	0.968300i	\(0.419638\pi\)
−0.963468	+	0.267824i	\(0.913696\pi\)
\(29\)	−119.004	−	206.121i	−0.762016	−	1.31985i	−0.941810	−	0.336146i	\(-0.890877\pi\)
							0.179794	−	0.983704i	\(-0.442457\pi\)
\(31\)	−37.5030	+	64.9570i	−0.217282	+	0.376343i	−0.953976	−	0.299883i	\(-0.903052\pi\)
							0.736694	+	0.676226i	\(0.236386\pi\)
\(37\)	18.8827			0.0839000			0.0419500	−	0.999120i	\(-0.486643\pi\)
							0.0419500	+	0.999120i	\(0.486643\pi\)
\(41\)	−85.4586	+	148.019i	−0.325522	+	0.563820i	−0.981618	−	0.190857i	\(-0.938873\pi\)
							0.656096	+	0.754677i	\(0.272207\pi\)
\(43\)	34.8186	+	60.3077i	0.123484	+	0.213880i	0.921139	−	0.389233i	\(-0.127260\pi\)
							−0.797656	+	0.603113i	\(0.793927\pi\)
\(47\)	254.788	+	441.305i	0.790736	+	1.36959i	0.925512	+	0.378719i	\(0.123635\pi\)
							−0.134776	+	0.990876i	\(0.543031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.4.e.a.58.19	✓	54
9.4	even	3		1539.4.a.h.1.9		27
9.5	odd	6		1539.4.a.g.1.19		27
9.7	even	3	inner	171.4.e.a.115.19	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.4.e.a.58.19	✓	54	1.1	even	1	trivial
171.4.e.a.115.19	yes	54	9.7	even	3	inner
1539.4.a.g.1.19		27	9.5	odd	6
1539.4.a.h.1.9		27	9.4	even	3