Embedded newform 189.2.e.a.163.1

• Label: 189.2.e.a.163.1
• Level: $189$
• Weight: $2$
• Character: 189.163
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	189.163
Dual form			189.2.e.a.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(2.00000 - 3.46410i) q^{5} +(2.00000 - 1.73205i) q^{7} -3.00000 q^{8} +(2.00000 + 3.46410i) q^{10} +(-1.00000 - 1.73205i) q^{11} +1.00000 q^{13} +(0.500000 + 2.59808i) q^{14} +(0.500000 - 0.866025i) q^{16} +(3.00000 + 5.19615i) q^{17} +(-2.00000 + 3.46410i) q^{19} +4.00000 q^{20} +2.00000 q^{22} +(-3.00000 + 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(-0.500000 + 0.866025i) q^{26} +(2.50000 + 0.866025i) q^{28} +2.00000 q^{29} +(-1.50000 - 2.59808i) q^{31} +(-2.50000 - 4.33013i) q^{32} -6.00000 q^{34} +(-2.00000 - 10.3923i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(-2.00000 - 3.46410i) q^{38} +(-6.00000 + 10.3923i) q^{40} -2.00000 q^{41} -1.00000 q^{43} +(1.00000 - 1.73205i) q^{44} +(-3.00000 - 5.19615i) q^{46} +(-3.00000 + 5.19615i) q^{47} +(1.00000 - 6.92820i) q^{49} +11.0000 q^{50} +(0.500000 + 0.866025i) q^{52} +(3.00000 + 5.19615i) q^{53} -8.00000 q^{55} +(-6.00000 + 5.19615i) q^{56} +(-1.00000 + 1.73205i) q^{58} +(-3.00000 - 5.19615i) q^{59} +(2.50000 - 4.33013i) q^{61} +3.00000 q^{62} +7.00000 q^{64} +(2.00000 - 3.46410i) q^{65} +(-3.50000 - 6.06218i) q^{67} +(-3.00000 + 5.19615i) q^{68} +(10.0000 + 3.46410i) q^{70} +(3.00000 + 5.19615i) q^{73} +(-1.50000 - 2.59808i) q^{74} -4.00000 q^{76} +(-5.00000 - 1.73205i) q^{77} +(-5.50000 + 9.52628i) q^{79} +(-2.00000 - 3.46410i) q^{80} +(1.00000 - 1.73205i) q^{82} +6.00000 q^{83} +24.0000 q^{85} +(0.500000 - 0.866025i) q^{86} +(3.00000 + 5.19615i) q^{88} +(2.00000 - 3.46410i) q^{89} +(2.00000 - 1.73205i) q^{91} -6.00000 q^{92} +(-3.00000 - 5.19615i) q^{94} +(8.00000 + 13.8564i) q^{95} +9.00000 q^{97} +(5.50000 + 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 + q^4 + 4 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^10 - 2 * q^11 + 2 * q^13 + q^14 + q^16 + 6 * q^17 - 4 * q^19 + 8 * q^20 + 4 * q^22 - 6 * q^23 - 11 * q^25 - q^26 + 5 * q^28 + 4 * q^29 - 3 * q^31 - 5 * q^32 - 12 * q^34 - 4 * q^35 - 3 * q^37 - 4 * q^38 - 12 * q^40 - 4 * q^41 - 2 * q^43 + 2 * q^44 - 6 * q^46 - 6 * q^47 + 2 * q^49 + 22 * q^50 + q^52 + 6 * q^53 - 16 * q^55 - 12 * q^56 - 2 * q^58 - 6 * q^59 + 5 * q^61 + 6 * q^62 + 14 * q^64 + 4 * q^65 - 7 * q^67 - 6 * q^68 + 20 * q^70 + 6 * q^73 - 3 * q^74 - 8 * q^76 - 10 * q^77 - 11 * q^79 - 4 * q^80 + 2 * q^82 + 12 * q^83 + 48 * q^85 + q^86 + 6 * q^88 + 4 * q^89 + 4 * q^91 - 12 * q^92 - 6 * q^94 + 16 * q^95 + 18 * q^97 + 11 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.500000	+	0.866025i	−0.353553	+	0.612372i	−0.986869	−	0.161521i	\(-0.948360\pi\)
							0.633316	+	0.773893i	\(0.281693\pi\)
\(3\)		0			0
							\(5\)	2.00000	−	3.46410i	0.894427	−	1.54919i	0.0599153	−	0.998203i	\(-0.480917\pi\)
0.834512	−	0.550990i	\(-0.185750\pi\)
\(7\)	2.00000	−	1.73205i	0.755929	−	0.654654i
							\(11\)	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	\(-0.264158\pi\)
−0.976478	+	0.215615i	\(0.930824\pi\)
\(13\)	1.00000			0.277350			0.138675	−	0.990338i	\(-0.455716\pi\)
							0.138675	+	0.990338i	\(0.455716\pi\)
\(17\)	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	\(0.0927008\pi\)
							−0.230285	+	0.973123i	\(0.573966\pi\)
\(19\)	−2.00000	+	3.46410i	−0.458831	+	0.794719i	−0.998899	−	0.0469020i	\(-0.985065\pi\)
							0.540068	+	0.841621i	\(0.318398\pi\)
\(23\)	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	\(0.381789\pi\)
							−0.988436	+	0.151642i	\(0.951544\pi\)
\(29\)	2.00000			0.371391			0.185695	−	0.982607i	\(-0.440546\pi\)
							0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	−1.50000	−	2.59808i	−0.269408	−	0.466628i	0.699301	−	0.714827i	\(-0.253495\pi\)
							−0.968709	+	0.248199i	\(0.920161\pi\)
\(37\)	−1.50000	+	2.59808i	−0.246598	+	0.427121i	−0.962580	−	0.270998i	\(-0.912646\pi\)
							0.715981	+	0.698119i	\(0.245980\pi\)
\(41\)	−2.00000			−0.312348			−0.156174	−	0.987730i	\(-0.549916\pi\)
							−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)	−1.00000			−0.152499			−0.0762493	−	0.997089i	\(-0.524294\pi\)
							−0.0762493	+	0.997089i	\(0.524294\pi\)
\(47\)	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	\(-0.977503\pi\)
							0.559908	+	0.828554i	\(0.310836\pi\)
\(53\)	3.00000	+	5.19615i	0.412082	+	0.713746i	0.995117	−	0.0987002i	\(-0.0314685\pi\)
							−0.583036	+	0.812447i	\(0.698135\pi\)
\(59\)	−3.00000	−	5.19615i	−0.390567	−	0.676481i	0.601958	−	0.798528i	\(-0.294388\pi\)
							−0.992524	+	0.122047i	\(0.961054\pi\)
\(61\)	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	\(-0.729619\pi\)
							0.980507	+	0.196485i	\(0.0629528\pi\)
\(67\)	−3.50000	−	6.06218i	−0.427593	−	0.740613i	0.569066	−	0.822292i	\(-0.307305\pi\)
							−0.996659	+	0.0816792i	\(0.973972\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	3.00000	+	5.19615i	0.351123	+	0.608164i	0.986447	−	0.164083i	\(-0.0524664\pi\)
							−0.635323	+	0.772246i	\(0.719133\pi\)
\(79\)	−5.50000	+	9.52628i	−0.618798	+	1.07179i	0.370907	+	0.928670i	\(0.379047\pi\)
							−0.989705	+	0.143120i	\(0.954286\pi\)
\(83\)	6.00000			0.658586			0.329293	−	0.944228i	\(-0.393190\pi\)
							0.329293	+	0.944228i	\(0.393190\pi\)
\(89\)	2.00000	−	3.46410i	0.212000	−	0.367194i	−0.740341	−	0.672232i	\(-0.765336\pi\)
							0.952340	+	0.305038i	\(0.0986691\pi\)
\(97\)	9.00000			0.913812			0.456906	−	0.889515i	\(-0.348958\pi\)
							0.456906	+	0.889515i	\(0.348958\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.e.a.163.1	yes	2
3.2	odd	2		189.2.e.c.163.1	yes	2
7.2	even	3		1323.2.a.m.1.1		1
7.4	even	3	inner	189.2.e.a.109.1	✓	2
7.5	odd	6		1323.2.a.p.1.1		1
9.2	odd	6		567.2.h.b.352.1		2
9.4	even	3		567.2.g.b.541.1		2
9.5	odd	6		567.2.g.e.541.1		2
9.7	even	3		567.2.h.e.352.1		2
21.2	odd	6		1323.2.a.g.1.1		1
21.5	even	6		1323.2.a.d.1.1		1
21.11	odd	6		189.2.e.c.109.1	yes	2
63.4	even	3		567.2.h.e.298.1		2
63.11	odd	6		567.2.g.e.109.1		2
63.25	even	3		567.2.g.b.109.1		2
63.32	odd	6		567.2.h.b.298.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.e.a.109.1	✓	2	7.4	even	3	inner
189.2.e.a.163.1	yes	2	1.1	even	1	trivial
189.2.e.c.109.1	yes	2	21.11	odd	6
189.2.e.c.163.1	yes	2	3.2	odd	2
567.2.g.b.109.1		2	63.25	even	3
567.2.g.b.541.1		2	9.4	even	3
567.2.g.e.109.1		2	63.11	odd	6
567.2.g.e.541.1		2	9.5	odd	6
567.2.h.b.298.1		2	63.32	odd	6
567.2.h.b.352.1		2	9.2	odd	6
567.2.h.e.298.1		2	63.4	even	3
567.2.h.e.352.1		2	9.7	even	3
1323.2.a.d.1.1		1	21.5	even	6
1323.2.a.g.1.1		1	21.2	odd	6
1323.2.a.m.1.1		1	7.2	even	3
1323.2.a.p.1.1		1	7.5	odd	6