Embedded newform 243.2.i.a.4.18

• Label: 243.2.i.a.4.18
• 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			4.18
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.08187 + 0.0419816i) q^{2} +(1.65295 - 0.517453i) q^{3} +(-0.825297 - 0.0641472i) q^{4} +(-2.72770 - 3.38181i) q^{5} +(1.81001 - 0.490425i) q^{6} +(3.01742 - 1.37158i) q^{7} +(-3.04091 - 0.355431i) q^{8} +(2.46448 - 1.71065i) q^{9} +(-2.80905 - 3.77321i) q^{10} +(0.0851812 + 4.39192i) q^{11} +(-1.39737 + 0.321020i) q^{12} +(1.85722 + 0.595493i) q^{13} +(3.32204 - 1.35720i) q^{14} +(-6.25867 - 4.17851i) q^{15} +(-1.63927 - 0.256377i) q^{16} +(0.778598 + 2.60070i) q^{17} +(2.73808 - 1.74724i) q^{18} +(3.54935 - 0.841212i) q^{19} +(2.03423 + 2.96597i) q^{20} +(4.27791 - 3.82853i) q^{21} +(-0.0922247 + 4.75508i) q^{22} +(-1.60031 + 1.14383i) q^{23} +(-5.21039 + 0.986017i) q^{24} +(-2.93781 + 13.5624i) q^{25} +(1.98427 + 0.722217i) q^{26} +(3.18849 - 4.10287i) q^{27} +(-2.57825 + 0.938406i) q^{28} +(0.476218 + 3.48653i) q^{29} +(-6.59567 - 4.78337i) q^{30} +(-4.48021 - 4.39416i) q^{31} +(4.24572 + 0.833833i) q^{32} +(2.41341 + 7.21555i) q^{33} +(0.733163 + 2.84631i) q^{34} +(-12.8690 - 6.46307i) q^{35} +(-2.14367 + 1.25370i) q^{36} +(2.05949 + 1.35455i) q^{37} +(3.87527 - 0.761078i) q^{38} +(3.37802 + 0.0232970i) q^{39} +(7.09267 + 11.2533i) q^{40} +(-0.0858964 + 0.163070i) q^{41} +(4.78888 - 3.96239i) q^{42} +(1.21942 - 3.56390i) q^{43} +(0.211430 - 3.63011i) q^{44} +(-12.5075 - 3.66830i) q^{45} +(-1.77935 + 1.17030i) q^{46} +(4.30163 - 4.21901i) q^{47} +(-2.84229 + 0.424466i) q^{48} +(2.62090 - 3.00322i) q^{49} +(-3.74771 + 14.5495i) q^{50} +(2.63272 + 3.89594i) q^{51} +(-1.49456 - 0.610594i) q^{52} +(-9.91480 - 8.31951i) q^{53} +(3.62179 - 4.30493i) q^{54} +(14.6203 - 12.2679i) q^{55} +(-9.66319 + 3.09838i) q^{56} +(5.43162 - 3.22711i) q^{57} +(0.368838 + 3.79198i) q^{58} +(-4.66733 + 2.57532i) q^{59} +(4.89723 + 3.84999i) q^{60} +(1.91579 - 0.148907i) q^{61} +(-4.66255 - 4.94201i) q^{62} +(5.09008 - 8.54198i) q^{63} +(7.78728 + 1.84562i) q^{64} +(-3.05208 - 7.90508i) q^{65} +(2.30809 + 7.90763i) q^{66} +(-0.815639 + 5.97153i) q^{67} +(-0.475747 - 2.19629i) q^{68} +(-2.05335 + 2.71878i) q^{69} +(-13.6513 - 7.53249i) q^{70} +(-4.41340 + 10.2314i) q^{71} +(-8.10229 + 4.32597i) q^{72} +(-2.40628 + 3.23220i) q^{73} +(2.17124 + 1.55191i) q^{74} +(2.16187 + 23.9382i) q^{75} +(-2.98323 + 0.466569i) q^{76} +(6.28091 + 13.1354i) q^{77} +(3.65362 + 0.167019i) q^{78} +(-2.54345 + 4.03546i) q^{79} +(3.60441 + 6.24302i) q^{80} +(3.14737 - 8.43173i) q^{81} +(-0.0997750 + 0.172815i) q^{82} +(-0.168282 - 0.319476i) q^{83} +(-3.77613 + 2.88526i) q^{84} +(6.67130 - 9.72699i) q^{85} +(1.46888 - 3.80450i) q^{86} +(2.59128 + 5.51665i) q^{87} +(1.30200 - 13.3857i) q^{88} +(6.14125 + 14.2370i) q^{89} +(-13.3775 - 4.49372i) q^{90} +(6.42076 - 0.750479i) q^{91} +(1.39410 - 0.841345i) q^{92} +(-9.67933 - 4.94503i) q^{93} +(4.83094 - 4.38384i) q^{94} +(-12.5264 - 9.70868i) q^{95} +(7.44944 - 0.818678i) q^{96} +(-6.93597 + 8.59926i) q^{97} +(2.96157 - 3.13908i) q^{98} +(7.72296 + 10.6781i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.08187	+	0.0419816i	0.765000	+	0.0296855i	0.418322	−	0.908299i	\(-0.362618\pi\)
							0.346678	+	0.937984i	\(0.387310\pi\)
\(3\)	1.65295	−	0.517453i	0.954331	−	0.298752i
							\(5\)	−2.72770	−	3.38181i	−1.21986	−	1.51239i	−0.794375	−	0.607428i	\(-0.792201\pi\)
−0.425488	−	0.904964i	\(-0.639897\pi\)
\(7\)	3.01742	−	1.37158i	1.14048	−	0.518410i	0.247667	−	0.968845i	\(-0.420336\pi\)
							0.892809	+	0.450435i	\(0.148731\pi\)
\(11\)	0.0851812	+	4.39192i	0.0256831	+	1.32421i	0.762000	+	0.647577i	\(0.224218\pi\)
							−0.736317	+	0.676637i	\(0.763437\pi\)
\(13\)	1.85722	+	0.595493i	0.515099	+	0.165160i	0.551496	−	0.834178i	\(-0.314057\pi\)
							−0.0363969	+	0.999337i	\(0.511588\pi\)
\(17\)	0.778598	+	2.60070i	0.188838	+	0.630762i	0.999006	+	0.0445653i	\(0.0141903\pi\)
							−0.810169	+	0.586197i	\(0.800625\pi\)
\(19\)	3.54935	−	0.841212i	0.814278	−	0.192987i	0.197681	−	0.980266i	\(-0.436659\pi\)
							0.616597	+	0.787279i	\(0.288511\pi\)
\(23\)	−1.60031	+	1.14383i	−0.333687	+	0.238505i	−0.735976	−	0.677008i	\(-0.763276\pi\)
							0.402289	+	0.915513i	\(0.368215\pi\)
\(29\)	0.476218	+	3.48653i	0.0884315	+	0.647433i	0.980641	+	0.195812i	\(0.0627343\pi\)
							−0.892210	+	0.451621i	\(0.850846\pi\)
\(31\)	−4.48021	−	4.39416i	−0.804670	−	0.789214i	0.176102	−	0.984372i	\(-0.443651\pi\)
							−0.980772	+	0.195158i	\(0.937478\pi\)
\(37\)	2.05949	+	1.35455i	0.338578	+	0.222686i	0.707400	−	0.706814i	\(-0.249868\pi\)
							−0.368821	+	0.929500i	\(0.620239\pi\)
\(41\)	−0.0858964	+	0.163070i	−0.0134148	+	0.0254673i	−0.891378	−	0.453262i	\(-0.850260\pi\)
							0.877963	+	0.478729i	\(0.158902\pi\)
\(43\)	1.21942	−	3.56390i	0.185961	−	0.543490i	−0.813304	−	0.581839i	\(-0.802333\pi\)
							0.999264	+	0.0383490i	\(0.0122099\pi\)
\(47\)	4.30163	−	4.21901i	0.627457	−	0.615406i	−0.315257	−	0.949006i	\(-0.602091\pi\)
							0.942714	+	0.333601i	\(0.108264\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.18	✓	1404
3.2	odd	2		729.2.i.a.64.9		1404
243.61	even	81	inner	243.2.i.a.61.18	yes	1404
243.182	odd	162		729.2.i.a.262.9		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.18	✓	1404	1.1	even	1	trivial
243.2.i.a.61.18	yes	1404	243.61	even	81	inner
729.2.i.a.64.9		1404	3.2	odd	2
729.2.i.a.262.9		1404	243.182	odd	162