Embedded newform 243.2.i.a.4.14

• Label: 243.2.i.a.4.14
• 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			4.14
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.242273 + 0.00940130i) q^{2} +(1.46469 + 0.924490i) q^{3} +(-1.93538 - 0.150429i) q^{4} +(1.25182 + 1.55201i) q^{5} +(0.346164 + 0.237749i) q^{6} +(1.52000 - 0.690924i) q^{7} +(-0.949108 - 0.110935i) q^{8} +(1.29064 + 2.70818i) q^{9} +(0.288691 + 0.387780i) q^{10} +(0.0181873 + 0.937731i) q^{11} +(-2.69566 - 2.00957i) q^{12} +(0.561299 + 0.179973i) q^{13} +(0.374750 - 0.153102i) q^{14} +(0.398709 + 3.43051i) q^{15} +(3.60690 + 0.564109i) q^{16} +(0.476732 + 1.59240i) q^{17} +(0.287226 + 0.668253i) q^{18} +(-6.11738 + 1.44984i) q^{19} +(-2.18928 - 3.19204i) q^{20} +(2.86508 + 0.393233i) q^{21} +(-0.00440960 + 0.227358i) q^{22} +(5.72988 - 4.09547i) q^{23} +(-1.28759 - 1.03993i) q^{24} +(0.216827 - 1.00098i) q^{25} +(0.134296 + 0.0488797i) q^{26} +(-0.613302 + 5.15983i) q^{27} +(-3.04571 + 1.10855i) q^{28} +(-0.509635 - 3.73119i) q^{29} +(0.0643451 + 0.834870i) q^{30} +(-6.77724 - 6.64707i) q^{31} +(2.74387 + 0.538878i) q^{32} +(-0.840284 + 1.39030i) q^{33} +(0.100529 + 0.390276i) q^{34} +(2.97509 + 1.49415i) q^{35} +(-2.09048 - 5.43551i) q^{36} +(-2.79500 - 1.83830i) q^{37} +(-1.49571 + 0.293747i) q^{38} +(0.655746 + 0.782521i) q^{39} +(-1.01594 - 1.61190i) q^{40} +(4.74187 - 9.00222i) q^{41} +(0.690435 + 0.122205i) q^{42} +(-0.529839 + 1.54851i) q^{43} +(0.105863 - 1.81760i) q^{44} +(-2.58749 + 5.39325i) q^{45} +(1.42670 - 0.938354i) q^{46} +(3.25586 - 3.19333i) q^{47} +(4.76148 + 4.16079i) q^{48} +(-2.76963 + 3.17364i) q^{49} +(0.0619419 - 0.240473i) q^{50} +(-0.773888 + 2.77310i) q^{51} +(-1.05925 - 0.432753i) q^{52} +(3.89454 + 3.26791i) q^{53} +(-0.197096 + 1.24432i) q^{54} +(-1.43260 + 1.20210i) q^{55} +(-1.51929 + 0.487141i) q^{56} +(-10.3004 - 3.53188i) q^{57} +(-0.0883928 - 0.908757i) q^{58} +(-5.05179 + 2.78746i) q^{59} +(-0.255602 - 6.69932i) q^{60} +(-2.51565 + 0.195532i) q^{61} +(-1.57945 - 1.67412i) q^{62} +(3.83292 + 3.22470i) q^{63} +(-6.44498 - 1.52749i) q^{64} +(0.423325 + 1.09644i) q^{65} +(-0.216649 + 0.328932i) q^{66} +(1.75171 - 12.8248i) q^{67} +(-0.683113 - 3.15360i) q^{68} +(12.1787 - 0.701384i) q^{69} +(0.706737 + 0.389961i) q^{70} +(-1.97332 + 4.57468i) q^{71} +(-0.924522 - 2.71353i) q^{72} +(-1.35027 + 1.81372i) q^{73} +(-0.659872 - 0.471648i) q^{74} +(1.24298 - 1.26568i) q^{75} +(12.0575 - 1.88576i) q^{76} +(0.675546 + 1.41278i) q^{77} +(0.151513 + 0.195749i) q^{78} +(-1.78312 + 2.82911i) q^{79} +(3.63969 + 6.30412i) q^{80} +(-5.66851 + 6.99057i) q^{81} +(1.23346 - 2.13642i) q^{82} +(-5.50994 - 10.4604i) q^{83} +(-5.48586 - 1.19205i) q^{84} +(-1.87464 + 2.73329i) q^{85} +(-0.142924 + 0.370182i) q^{86} +(2.70299 - 5.93619i) q^{87} +(0.0867653 - 0.892025i) q^{88} +(-0.0128353 - 0.0297555i) q^{89} +(-0.677583 + 1.28231i) q^{90} +(0.977523 - 0.114256i) q^{91} +(-11.7056 + 7.06434i) q^{92} +(-3.78141 - 16.0014i) q^{93} +(0.818829 - 0.743048i) q^{94} +(-9.90803 - 7.67931i) q^{95} +(3.52073 + 3.32597i) q^{96} +(1.23205 - 1.52750i) q^{97} +(-0.700843 + 0.742850i) q^{98} +(-2.51607 + 1.25953i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.242273	+	0.00940130i	0.171313	+	0.00664772i	0.124289	−	0.992246i	\(-0.460335\pi\)
							0.0470238	+	0.998894i	\(0.485026\pi\)
\(3\)	1.46469	+	0.924490i	0.845640	+	0.533754i
							\(5\)	1.25182	+	1.55201i	0.559831	+	0.694082i	0.976633	−	0.214916i	\(-0.0689477\pi\)
−0.416802	+	0.908998i	\(0.636849\pi\)
\(7\)	1.52000	−	0.690924i	0.574506	−	0.261145i	−0.105418	−	0.994428i	\(-0.533618\pi\)
							0.679923	+	0.733283i	\(0.262013\pi\)
\(11\)	0.0181873	+	0.937731i	0.00548367	+	0.282736i	0.992509	+	0.122172i	\(0.0389859\pi\)
							−0.987025	+	0.160565i	\(0.948668\pi\)
\(13\)	0.561299	+	0.179973i	0.155676	+	0.0499156i	0.382142	−	0.924103i	\(-0.375186\pi\)
							−0.226466	+	0.974019i	\(0.572717\pi\)
\(17\)	0.476732	+	1.59240i	0.115624	+	0.386213i	0.996137	−	0.0878098i	\(-0.0279868\pi\)
							−0.880513	+	0.474022i	\(0.842802\pi\)
\(19\)	−6.11738	+	1.44984i	−1.40342	+	0.332617i	−0.861477	−	0.507797i	\(-0.830460\pi\)
							−0.541945	+	0.840414i	\(0.682312\pi\)
\(23\)	5.72988	−	4.09547i	1.19476	−	0.853965i	0.202642	−	0.979253i	\(-0.435047\pi\)
							0.992120	+	0.125288i	\(0.0399856\pi\)
\(29\)	−0.509635	−	3.73119i	−0.0946368	−	0.692864i	−0.975580	−	0.219643i	\(-0.929511\pi\)
							0.880944	−	0.473221i	\(-0.156909\pi\)
\(31\)	−6.77724	−	6.64707i	−1.21723	−	1.19385i	−0.974654	−	0.223716i	\(-0.928181\pi\)
							−0.242573	−	0.970133i	\(-0.577991\pi\)
\(37\)	−2.79500	−	1.83830i	−0.459496	−	0.302215i	0.298576	−	0.954386i	\(-0.403488\pi\)
							−0.758072	+	0.652171i	\(0.773859\pi\)
\(41\)	4.74187	−	9.00222i	0.740556	−	1.40591i	−0.166990	−	0.985959i	\(-0.553405\pi\)
							0.907546	−	0.419952i	\(-0.137953\pi\)
\(43\)	−0.529839	+	1.54851i	−0.0807997	+	0.236146i	−0.979152	−	0.203130i	\(-0.934889\pi\)
							0.898352	+	0.439276i	\(0.144765\pi\)
\(47\)	3.25586	−	3.19333i	0.474916	−	0.465795i	−0.422893	−	0.906180i	\(-0.638985\pi\)
							0.897809	+	0.440385i	\(0.145158\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.14	✓	1404
3.2	odd	2		729.2.i.a.64.13		1404
243.61	even	81	inner	243.2.i.a.61.14	yes	1404
243.182	odd	162		729.2.i.a.262.13		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.14	✓	1404	1.1	even	1	trivial
243.2.i.a.61.14	yes	1404	243.61	even	81	inner
729.2.i.a.64.13		1404	3.2	odd	2
729.2.i.a.262.13		1404	243.182	odd	162