Embedded newform 189.2.bd.a.173.6

• Label: 189.2.bd.a.173.6
• Level: $189$
• Weight: $2$
• Character: 189.173
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.bd (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			173.6
Character	\(\chi\)	\(=\)	189.173
Dual form			189.2.bd.a.59.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.600445 - 1.64971i) q^{2} +(1.26150 - 1.18685i) q^{3} +(-0.828916 + 0.695543i) q^{4} +(0.270023 - 0.226576i) q^{5} +(-2.71542 - 1.36847i) q^{6} +(2.53352 + 0.762430i) q^{7} +(-1.39560 - 0.805749i) q^{8} +(0.182767 - 2.99443i) q^{9} +(-0.535918 - 0.309412i) q^{10} +(-0.647450 + 0.771601i) q^{11} +(-0.220171 + 1.86123i) q^{12} +(0.216492 + 0.258005i) q^{13} +(-0.263449 - 4.63736i) q^{14} +(0.0717217 - 0.606302i) q^{15} +(-0.867071 + 4.91741i) q^{16} +(-2.61130 + 4.52291i) q^{17} +(-5.04967 + 1.49648i) q^{18} +(-0.480077 + 0.277173i) q^{19} +(-0.0662328 + 0.375625i) q^{20} +(4.10092 - 2.04510i) q^{21} +(1.66167 + 0.604800i) q^{22} +(0.811938 - 2.23078i) q^{23} +(-2.71685 + 0.639915i) q^{24} +(-0.846665 + 4.80168i) q^{25} +(0.295641 - 0.512066i) q^{26} +(-3.32338 - 3.99439i) q^{27} +(-2.63037 + 1.13018i) q^{28} +(0.750003 - 0.893819i) q^{29} +(-1.04329 + 0.245731i) q^{30} +(1.12246 + 1.33769i) q^{31} +(5.45888 - 0.962549i) q^{32} +(0.0990172 + 1.74180i) q^{33} +(9.02943 + 1.59213i) q^{34} +(0.856855 - 0.368160i) q^{35} +(1.93126 + 2.60925i) q^{36} -6.14160 q^{37} +(0.745514 + 0.625561i) q^{38} +(0.579318 + 0.0685296i) q^{39} +(-0.559406 + 0.0986384i) q^{40} +(8.82053 - 7.40131i) q^{41} +(-5.83620 - 5.53736i) q^{42} +(5.78827 - 2.10676i) q^{43} -1.08992i q^{44} +(-0.629114 - 0.849974i) q^{45} -4.16766 q^{46} +(4.23097 + 3.55021i) q^{47} +(4.74242 + 7.23239i) q^{48} +(5.83740 + 3.86326i) q^{49} +(8.42975 - 1.48639i) q^{50} +(2.07386 + 8.80488i) q^{51} +(-0.358907 - 0.0632850i) q^{52} +(-3.63746 + 2.10009i) q^{53} +(-4.59407 + 7.88102i) q^{54} +0.355046i q^{55} +(-2.92144 - 3.10542i) q^{56} +(-0.276655 + 0.919434i) q^{57} +(-1.92488 - 0.700598i) q^{58} +(2.30113 + 13.0504i) q^{59} +(0.362258 + 0.552459i) q^{60} +(9.07139 - 10.8109i) q^{61} +(1.53283 - 2.65494i) q^{62} +(2.74608 - 7.44708i) q^{63} +(0.127580 + 0.220974i) q^{64} +(0.116915 + 0.0206153i) q^{65} +(2.81401 - 1.20921i) q^{66} +(-7.87228 - 2.86527i) q^{67} +(-0.981328 - 5.56539i) q^{68} +(-1.62335 - 3.77778i) q^{69} +(-1.12185 - 1.19250i) q^{70} +(-9.96202 + 5.75157i) q^{71} +(-2.66782 + 4.03175i) q^{72} -7.95615i q^{73} +(3.68770 + 10.1319i) q^{74} +(4.63081 + 7.06218i) q^{75} +(0.205158 - 0.563667i) q^{76} +(-2.22862 + 1.46123i) q^{77} +(-0.234794 - 0.996854i) q^{78} +(-11.1407 + 4.05487i) q^{79} +(0.880037 + 1.52427i) q^{80} +(-8.93319 - 1.09456i) q^{81} +(-17.5062 - 10.1072i) q^{82} +(-2.81649 - 2.36332i) q^{83} +(-1.97686 + 4.54759i) q^{84} +(0.319671 + 1.81295i) q^{85} +(-6.95108 - 8.28397i) q^{86} +(-0.114701 - 2.01770i) q^{87} +(1.52530 - 0.555162i) q^{88} +(1.99527 + 3.45591i) q^{89} +(-1.02446 + 1.54822i) q^{90} +(0.351774 + 0.818719i) q^{91} +(0.878576 + 2.41387i) q^{92} +(3.00362 + 0.355309i) q^{93} +(3.31634 - 9.11158i) q^{94} +(-0.0668311 + 0.183617i) q^{95} +(5.74398 - 7.69314i) q^{96} +(-5.45431 - 14.9856i) q^{97} +(2.86821 - 11.9497i) q^{98} +(2.19217 + 2.07976i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	132 * q - 3 * q^2 - 9 * q^3 - 3 * q^4 - 9 * q^5 + 18 * q^6 - 6 * q^7 - 18 * q^8 - 15 * q^9 - 9 * q^10 + 9 * q^11 - 9 * q^12 - 42 * q^14 - 24 * q^15 - 15 * q^16 - 9 * q^17 - 3 * q^18 - 9 * q^19 - 18 * q^20 + 15 * q^21 - 12 * q^22 + 30 * q^23 - 36 * q^24 - 3 * q^25 - 12 * q^28 + 6 * q^29 - 3 * q^30 - 9 * q^31 - 51 * q^32 - 9 * q^33 + 18 * q^34 - 9 * q^35 - 6 * q^37 - 9 * q^38 - 9 * q^39 - 9 * q^40 + 27 * q^42 - 12 * q^43 - 63 * q^45 - 6 * q^46 + 45 * q^47 + 30 * q^49 - 9 * q^50 + 33 * q^51 - 9 * q^52 + 45 * q^53 + 117 * q^54 - 51 * q^56 - 3 * q^58 - 9 * q^59 - 15 * q^60 - 63 * q^61 + 99 * q^62 - 33 * q^63 + 18 * q^64 - 102 * q^65 + 63 * q^66 - 3 * q^67 + 144 * q^68 - 108 * q^69 - 15 * q^70 + 18 * q^71 + 15 * q^72 - 33 * q^74 - 9 * q^75 - 36 * q^76 - 57 * q^77 + 66 * q^78 - 21 * q^79 - 72 * q^80 + 57 * q^81 - 18 * q^82 + 90 * q^83 + 51 * q^84 + 9 * q^85 - 33 * q^86 - 9 * q^87 + 45 * q^88 - 9 * q^89 - 81 * q^90 - 21 * q^91 + 150 * q^92 - 87 * q^93 - 9 * q^94 + 27 * q^95 - 9 * q^96 - 180 * q^98 + 96 * q^99

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.600445	−	1.64971i	−0.424579	−	1.16652i	−0.949059	−	0.315098i	\(-0.897963\pi\)
							0.524481	−	0.851422i	\(-0.324260\pi\)
\(3\)	1.26150	−	1.18685i	0.728328	−	0.685229i
							\(5\)	0.270023	−	0.226576i	0.120758	−	0.101328i	−0.580409	−	0.814325i	\(-0.697107\pi\)
0.701167	+	0.712997i	\(0.252663\pi\)
\(7\)	2.53352	+	0.762430i	0.957579	+	0.288171i
							\(11\)	−0.647450	+	0.771601i	−0.195213	+	0.232646i	−0.854768	−	0.519010i	\(-0.826301\pi\)
0.659554	+	0.751657i	\(0.270745\pi\)
\(13\)	0.216492	+	0.258005i	0.0600440	+	0.0715576i	0.795230	−	0.606307i	\(-0.207350\pi\)
							−0.735186	+	0.677865i	\(0.762905\pi\)
\(17\)	−2.61130	+	4.52291i	−0.633334	+	1.09697i	0.353531	+	0.935423i	\(0.384981\pi\)
							−0.986865	+	0.161544i	\(0.948353\pi\)
\(19\)	−0.480077	+	0.277173i	−0.110137	+	0.0635878i	−0.554057	−	0.832479i	\(-0.686921\pi\)
							0.443920	+	0.896067i	\(0.353588\pi\)
\(23\)	0.811938	−	2.23078i	0.169301	−	0.465150i	−0.825806	−	0.563954i	\(-0.809280\pi\)
							0.995107	+	0.0988041i	\(0.0315017\pi\)
\(29\)	0.750003	−	0.893819i	0.139272	−	0.165978i	−0.691900	−	0.721994i	\(-0.743226\pi\)
							0.831172	+	0.556015i	\(0.187670\pi\)
\(31\)	1.12246	+	1.33769i	0.201599	+	0.240257i	0.857366	−	0.514707i	\(-0.172099\pi\)
							−0.655767	+	0.754963i	\(0.727655\pi\)
\(37\)	−6.14160			−1.00967			−0.504837	−	0.863215i	\(-0.668447\pi\)
							−0.504837	+	0.863215i	\(0.668447\pi\)
\(41\)	8.82053	−	7.40131i	1.37754	−	1.15589i	0.407422	−	0.913240i	\(-0.366428\pi\)
							0.970114	−	0.242650i	\(-0.0780166\pi\)
\(43\)	5.78827	−	2.10676i	0.882703	−	0.321278i	0.139403	−	0.990236i	\(-0.455482\pi\)
							0.743300	+	0.668958i	\(0.233259\pi\)
\(47\)	4.23097	+	3.55021i	0.617151	+	0.517851i	0.896906	−	0.442220i	\(-0.145809\pi\)
							−0.279756	+	0.960071i	\(0.590253\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.bd.a.173.6	yes	132
3.2	odd	2		567.2.bd.a.278.17		132
7.3	odd	6		189.2.ba.a.38.17	yes	132
21.17	even	6		567.2.ba.a.521.6		132
27.5	odd	18		189.2.ba.a.5.17	✓	132
27.22	even	9		567.2.ba.a.530.6		132
189.59	even	18	inner	189.2.bd.a.59.6	yes	132
189.157	odd	18		567.2.bd.a.206.17		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.5.17	✓	132	27.5	odd	18
189.2.ba.a.38.17	yes	132	7.3	odd	6
189.2.bd.a.59.6	yes	132	189.59	even	18	inner
189.2.bd.a.173.6	yes	132	1.1	even	1	trivial
567.2.ba.a.521.6		132	21.17	even	6
567.2.ba.a.530.6		132	27.22	even	9
567.2.bd.a.206.17		132	189.157	odd	18
567.2.bd.a.278.17		132	3.2	odd	2