Embedded newform 243.2.i.a.4.1

• Label: 243.2.i.a.4.1
• Level: $243$
• Weight: $2$
• Character: 243.4
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $1404$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.i (of order \(81\), degree \(54\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94036476912\)
Analytic rank:	\(0\)
Dimension:	\(1404\)
Relative dimension:	\(26\) over \(\Q(\zeta_{81})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{81}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.78387 - 0.108027i) q^{2} +(-1.71620 + 0.233793i) q^{3} +(5.74428 + 0.446480i) q^{4} +(2.42193 + 3.00272i) q^{5} +(4.80293 - 0.465452i) q^{6} +(0.433636 - 0.197112i) q^{7} +(-10.4088 - 1.21662i) q^{8} +(2.89068 - 0.802469i) q^{9} +(-6.41795 - 8.62081i) q^{10} +(-0.0351445 - 1.81204i) q^{11} +(-9.96271 + 0.576719i) q^{12} +(1.12662 + 0.361236i) q^{13} +(-1.22848 + 0.501889i) q^{14} +(-4.85852 - 4.58703i) q^{15} +(17.4606 + 2.73079i) q^{16} +(0.970474 + 3.24161i) q^{17} +(-8.13397 + 1.92170i) q^{18} +(-0.623466 + 0.147764i) q^{19} +(12.5716 + 18.3298i) q^{20} +(-0.698122 + 0.439664i) q^{21} +(-0.0979114 + 5.04829i) q^{22} +(3.59136 - 2.56695i) q^{23} +(18.1481 - 0.345548i) q^{24} +(-2.09206 + 9.65804i) q^{25} +(-3.09734 - 1.12734i) q^{26} +(-4.77338 + 2.05302i) q^{27} +(2.57893 - 0.938654i) q^{28} +(0.389330 + 2.85040i) q^{29} +(13.0300 + 13.2946i) q^{30} +(2.65887 + 2.60780i) q^{31} +(-27.7465 - 5.44924i) q^{32} +(0.483957 + 3.10161i) q^{33} +(-2.35149 - 9.12906i) q^{34} +(1.64210 + 0.824696i) q^{35} +(16.9632 - 3.31897i) q^{36} +(0.223578 + 0.147050i) q^{37} +(1.75161 - 0.344005i) q^{38} +(-2.01796 - 0.356558i) q^{39} +(-21.5563 - 34.2013i) q^{40} +(-0.610808 + 1.15959i) q^{41} +(1.99098 - 1.14855i) q^{42} +(-2.53192 + 7.39980i) q^{43} +(0.607162 - 10.4246i) q^{44} +(9.41061 + 6.73638i) q^{45} +(-10.2752 + 6.75809i) q^{46} +(3.10099 - 3.04143i) q^{47} +(-30.6043 - 0.604419i) q^{48} +(-4.45346 + 5.10310i) q^{49} +(6.86736 - 26.6607i) q^{50} +(-2.42339 - 5.33636i) q^{51} +(6.31033 + 2.57805i) q^{52} +(2.01599 + 1.69162i) q^{53} +(13.5102 - 5.19968i) q^{54} +(5.35593 - 4.49416i) q^{55} +(-4.75345 + 1.52413i) q^{56} +(1.03545 - 0.399355i) q^{57} +(-0.775924 - 7.97719i) q^{58} +(5.69963 - 3.14493i) q^{59} +(-25.8607 - 28.5184i) q^{60} +(-8.20879 + 0.638038i) q^{61} +(-7.12024 - 7.54701i) q^{62} +(1.09533 - 0.917766i) q^{63} +(42.2611 + 10.0161i) q^{64} +(1.64390 + 4.25780i) q^{65} +(-1.01222 - 8.68676i) q^{66} +(-1.58041 + 11.5706i) q^{67} +(4.12736 + 19.0540i) q^{68} +(-5.56336 + 5.24503i) q^{69} +(-4.48231 - 2.47324i) q^{70} +(-3.08168 + 7.14413i) q^{71} +(-31.0649 + 4.83591i) q^{72} +(6.32253 - 8.49263i) q^{73} +(-0.606527 - 0.433519i) q^{74} +(1.33242 - 17.0642i) q^{75} +(-3.64734 + 0.570433i) q^{76} +(-0.372415 - 0.778839i) q^{77} +(5.57921 + 1.21060i) q^{78} +(-3.50813 + 5.56602i) q^{79} +(34.0885 + 59.0430i) q^{80} +(7.71209 - 4.63937i) q^{81} +(1.82568 - 3.16217i) q^{82} +(-4.12067 - 7.82290i) q^{83} +(-4.20651 + 2.21385i) q^{84} +(-7.38322 + 10.7650i) q^{85} +(7.84790 - 20.3266i) q^{86} +(-1.33457 - 4.80083i) q^{87} +(-1.83875 + 18.9040i) q^{88} +(-3.88578 - 9.00824i) q^{89} +(-25.4702 - 19.7698i) q^{90} +(0.559746 - 0.0654249i) q^{91} +(21.7759 - 13.1418i) q^{92} +(-5.17284 - 3.85389i) q^{93} +(-8.96132 + 8.13196i) q^{94} +(-1.95368 - 1.51422i) q^{95} +(48.8926 + 2.86505i) q^{96} +(10.8281 - 13.4247i) q^{97} +(12.9491 - 13.7253i) q^{98} +(-1.55570 - 5.20984i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	1404 * q - 54 * q^2 - 54 * q^3 - 54 * q^4 - 54 * q^5 - 54 * q^6 - 54 * q^7 - 54 * q^8 - 54 * q^9 - 54 * q^10 - 54 * q^11 - 54 * q^12 - 54 * q^13 - 54 * q^14 - 54 * q^15 - 54 * q^16 - 54 * q^17 - 54 * q^18 - 54 * q^19 - 54 * q^20 - 54 * q^21 - 54 * q^22 - 54 * q^23 - 54 * q^24 - 54 * q^25 - 54 * q^26 - 54 * q^27 - 54 * q^28 - 54 * q^29 - 54 * q^30 - 54 * q^31 - 54 * q^32 - 54 * q^33 - 54 * q^34 - 54 * q^35 - 54 * q^36 - 54 * q^37 - 54 * q^38 - 54 * q^39 - 54 * q^40 - 54 * q^41 - 54 * q^42 - 54 * q^43 - 54 * q^44 - 54 * q^45 - 54 * q^46 - 54 * q^47 - 54 * q^48 - 54 * q^49 - 54 * q^50 - 54 * q^51 - 54 * q^52 - 54 * q^53 - 54 * q^54 - 54 * q^55 - 54 * q^56 - 54 * q^57 - 54 * q^58 - 54 * q^59 - 54 * q^60 - 54 * q^61 - 54 * q^62 - 54 * q^63 - 54 * q^64 + 54 * q^66 - 54 * q^67 + 135 * q^68 + 54 * q^69 - 54 * q^70 + 54 * q^71 + 270 * q^72 - 54 * q^73 + 162 * q^74 + 81 * q^75 - 54 * q^76 + 162 * q^77 + 162 * q^78 - 54 * q^79 + 351 * q^80 + 54 * q^81 - 27 * q^82 + 54 * q^83 + 324 * q^84 - 54 * q^85 + 162 * q^86 + 162 * q^87 - 54 * q^88 + 81 * q^89 + 162 * q^90 - 54 * q^91 + 270 * q^92 + 54 * q^93 - 54 * q^94 + 54 * q^95 + 135 * q^96 - 54 * q^97 + 54 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{81}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.78387	−	0.108027i	−1.96849	−	0.0763865i	−0.977553	−	0.210690i	\(-0.932429\pi\)
							−0.990940	+	0.134303i	\(0.957120\pi\)
\(3\)	−1.71620	+	0.233793i	−0.990848	+	0.134980i
							\(5\)	2.42193	+	3.00272i	1.08312	+	1.34286i	0.935306	+	0.353840i	\(0.115124\pi\)
0.147812	+	0.989015i	\(0.452777\pi\)
\(7\)	0.433636	−	0.197112i	0.163899	−	0.0745012i	−0.330181	−	0.943918i	\(-0.607110\pi\)
							0.494080	+	0.869416i	\(0.335505\pi\)
\(11\)	−0.0351445	−	1.81204i	−0.0105965	−	0.546351i	−0.967067	−	0.254522i	\(-0.918082\pi\)
							0.956470	−	0.291829i	\(-0.0942639\pi\)
\(13\)	1.12662	+	0.361236i	0.312468	+	0.100189i	0.457422	−	0.889250i	\(-0.348773\pi\)
							−0.144955	+	0.989438i	\(0.546304\pi\)
\(17\)	0.970474	+	3.24161i	0.235375	+	0.786206i	0.991308	+	0.131558i	\(0.0419979\pi\)
							−0.755934	+	0.654648i	\(0.772817\pi\)
\(19\)	−0.623466	+	0.147764i	−0.143033	+	0.0338994i	−0.301509	−	0.953463i	\(-0.597490\pi\)
							0.158476	+	0.987363i	\(0.449342\pi\)
\(23\)	3.59136	−	2.56695i	0.748850	−	0.535246i	−0.141822	−	0.989892i	\(-0.545296\pi\)
							0.890672	+	0.454646i	\(0.150234\pi\)
\(29\)	0.389330	+	2.85040i	0.0722967	+	0.529305i	0.990949	+	0.134238i	\(0.0428588\pi\)
							−0.918652	+	0.395067i	\(0.870721\pi\)
\(31\)	2.65887	+	2.60780i	0.477547	+	0.468375i	0.898668	−	0.438630i	\(-0.144536\pi\)
							−0.421121	+	0.907005i	\(0.638363\pi\)
\(37\)	0.223578	+	0.147050i	0.0367560	+	0.0241748i	0.567754	−	0.823198i	\(-0.307812\pi\)
							−0.530998	+	0.847373i	\(0.678183\pi\)
\(41\)	−0.610808	+	1.15959i	−0.0953922	+	0.181098i	−0.927812	−	0.373048i	\(-0.878313\pi\)
							0.832420	+	0.554145i	\(0.186955\pi\)
\(43\)	−2.53192	+	7.39980i	−0.386113	+	1.12846i	0.566444	+	0.824100i	\(0.308319\pi\)
							−0.952557	+	0.304359i	\(0.901558\pi\)
\(47\)	3.10099	−	3.04143i	0.452326	−	0.443639i	−0.437991	−	0.898979i	\(-0.644310\pi\)
							0.890317	+	0.455341i	\(0.150483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.1	✓	1404
3.2	odd	2		729.2.i.a.64.26		1404
243.61	even	81	inner	243.2.i.a.61.1	yes	1404
243.182	odd	162		729.2.i.a.262.26		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.1	✓	1404	1.1	even	1	trivial
243.2.i.a.61.1	yes	1404	243.61	even	81	inner
729.2.i.a.64.26		1404	3.2	odd	2
729.2.i.a.262.26		1404	243.182	odd	162