Embedded newform 243.2.i.a.4.13

• Label: 243.2.i.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0783119 + 0.00303886i) q^{2} +(-0.825726 - 1.52256i) q^{3} +(-1.98786 - 0.154509i) q^{4} +(2.71771 + 3.36944i) q^{5} +(-0.0600373 - 0.121743i) q^{6} +(-3.85378 + 1.75176i) q^{7} +(-0.310885 - 0.0363373i) q^{8} +(-1.63635 + 2.51443i) q^{9} +(0.202590 + 0.272126i) q^{10} +(0.0195362 + 1.00728i) q^{11} +(1.40618 + 3.15421i) q^{12} +(3.42761 + 1.09902i) q^{13} +(-0.307120 + 0.125473i) q^{14} +(2.88607 - 6.92010i) q^{15} +(3.91559 + 0.612387i) q^{16} +(0.293146 + 0.979175i) q^{17} +(-0.135787 + 0.191937i) q^{18} +(-2.47221 + 0.585924i) q^{19} +(-4.88183 - 7.11789i) q^{20} +(5.84932 + 4.42113i) q^{21} +(-0.00153107 + 0.0789416i) q^{22} +(-3.99963 + 2.85876i) q^{23} +(0.201381 + 0.503345i) q^{24} +(-2.90862 + 13.4277i) q^{25} +(0.265083 + 0.0964821i) q^{26} +(5.17954 + 0.415212i) q^{27} +(7.93146 - 2.88681i) q^{28} +(-0.937257 - 6.86194i) q^{29} +(0.247043 - 0.533156i) q^{30} +(1.79617 + 1.76167i) q^{31} +(0.919046 + 0.180495i) q^{32} +(1.51751 - 0.861484i) q^{33} +(0.0199812 + 0.0775718i) q^{34} +(-16.3759 - 8.22430i) q^{35} +(3.64135 - 4.74551i) q^{36} +(-3.78539 - 2.48969i) q^{37} +(-0.195384 + 0.0383721i) q^{38} +(-1.15695 - 6.12621i) q^{39} +(-0.722462 - 1.14626i) q^{40} +(-2.10893 + 4.00370i) q^{41} +(0.444636 + 0.364002i) q^{42} +(0.183220 - 0.535481i) q^{43} +(0.116799 - 2.00536i) q^{44} +(-12.9193 + 1.31991i) q^{45} +(-0.321906 + 0.211721i) q^{46} +(2.00535 - 1.96684i) q^{47} +(-2.30081 - 6.46736i) q^{48} +(7.18034 - 8.22776i) q^{49} +(-0.268584 + 1.04271i) q^{50} +(1.24879 - 1.25486i) q^{51} +(-6.64380 - 2.71429i) q^{52} +(-1.78492 - 1.49773i) q^{53} +(0.404357 + 0.0482559i) q^{54} +(-3.34088 + 2.80333i) q^{55} +(1.26174 - 0.404561i) q^{56} +(2.93347 + 3.28026i) q^{57} +(-0.0525459 - 0.540219i) q^{58} +(8.23503 - 4.54389i) q^{59} +(-6.80633 + 13.3103i) q^{60} +(11.9013 - 0.925042i) q^{61} +(0.135308 + 0.143418i) q^{62} +(1.90148 - 12.5566i) q^{63} +(-7.64129 - 1.81102i) q^{64} +(5.61218 + 14.5359i) q^{65} +(0.121457 - 0.0628529i) q^{66} +(-0.918505 + 6.72465i) q^{67} +(-0.431442 - 1.99176i) q^{68} +(7.65522 + 3.72910i) q^{69} +(-1.25744 - 0.693825i) q^{70} +(1.12709 - 2.61288i) q^{71} +(0.600086 - 0.722238i) q^{72} +(-3.95580 + 5.31356i) q^{73} +(-0.288875 - 0.206476i) q^{74} +(22.8461 - 6.65904i) q^{75} +(5.00494 - 0.782758i) q^{76} +(-1.83981 - 3.84763i) q^{77} +(-0.0719860 - 0.483271i) q^{78} +(-4.34181 + 6.88874i) q^{79} +(8.57805 + 14.8576i) q^{80} +(-3.64469 - 8.22899i) q^{81} +(-0.177321 + 0.307128i) q^{82} +(1.39686 + 2.65188i) q^{83} +(-10.9445 - 9.69237i) q^{84} +(-2.50258 + 3.64885i) q^{85} +(0.0159755 - 0.0413777i) q^{86} +(-9.67377 + 7.09311i) q^{87} +(0.0305284 - 0.313859i) q^{88} +(3.85283 + 8.93188i) q^{89} +(-1.01575 + 0.0641042i) q^{90} +(-15.1345 + 1.76897i) q^{91} +(8.39241 - 5.06485i) q^{92} +(1.19910 - 4.18942i) q^{93} +(0.163020 - 0.147933i) q^{94} +(-8.69299 - 6.73757i) q^{95} +(-0.484066 - 1.54834i) q^{96} +(5.53700 - 6.86481i) q^{97} +(0.587309 - 0.622511i) q^{98} +(-2.56471 - 1.59915i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0783119	+	0.00303886i	0.0553749	+	0.00214880i	0.0664479	−	0.997790i	\(-0.478833\pi\)
							−0.0110731	+	0.999939i	\(0.503525\pi\)
\(3\)	−0.825726	−	1.52256i	−0.476733	−	0.879048i
							\(5\)	2.71771	+	3.36944i	1.21540	+	1.50686i	0.803139	+	0.595792i	\(0.203162\pi\)
0.412260	+	0.911066i	\(0.364739\pi\)
\(7\)	−3.85378	+	1.75176i	−1.45659	+	0.662103i	−0.976579	−	0.215158i	\(-0.930974\pi\)
							−0.480014	+	0.877261i	\(0.659369\pi\)
\(11\)	0.0195362	+	1.00728i	0.00589039	+	0.303707i	0.991158	+	0.132687i	\(0.0423606\pi\)
							−0.985268	+	0.171020i	\(0.945294\pi\)
\(13\)	3.42761	+	1.09902i	0.950647	+	0.304813i	0.739892	−	0.672726i	\(-0.234877\pi\)
							0.210755	+	0.977539i	\(0.432408\pi\)
\(17\)	0.293146	+	0.979175i	0.0710983	+	0.237485i	0.986151	−	0.165849i	\(-0.0530365\pi\)
							−0.915053	+	0.403334i	\(0.867851\pi\)
\(19\)	−2.47221	+	0.585924i	−0.567163	+	0.134420i	−0.504186	−	0.863595i	\(-0.668208\pi\)
							−0.0629768	+	0.998015i	\(0.520059\pi\)
\(23\)	−3.99963	+	2.85876i	−0.833980	+	0.596093i	−0.916291	−	0.400514i	\(-0.868832\pi\)
							0.0823109	+	0.996607i	\(0.473770\pi\)
\(29\)	−0.937257	−	6.86194i	−0.174044	−	1.27423i	−0.845817	−	0.533473i	\(-0.820887\pi\)
							0.671773	−	0.740757i	\(-0.265533\pi\)
\(31\)	1.79617	+	1.76167i	0.322602	+	0.316405i	0.843412	−	0.537267i	\(-0.180543\pi\)
							−0.520811	+	0.853672i	\(0.674370\pi\)
\(37\)	−3.78539	−	2.48969i	−0.622314	−	0.409303i	0.198821	−	0.980036i	\(-0.436289\pi\)
							−0.821136	+	0.570733i	\(0.806659\pi\)
\(41\)	−2.10893	+	4.00370i	−0.329359	+	0.625273i	−0.992364	−	0.123342i	\(-0.960639\pi\)
							0.663005	+	0.748615i	\(0.269281\pi\)
\(43\)	0.183220	−	0.535481i	0.0279408	−	0.0816600i	−0.931296	−	0.364262i	\(-0.881321\pi\)
							0.959237	+	0.282602i	\(0.0911977\pi\)
\(47\)	2.00535	−	1.96684i	0.292511	−	0.286893i	−0.539111	−	0.842235i	\(-0.681240\pi\)
							0.831622	+	0.555342i	\(0.187413\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.13	✓	1404
3.2	odd	2		729.2.i.a.64.14		1404
243.61	even	81	inner	243.2.i.a.61.13	yes	1404
243.182	odd	162		729.2.i.a.262.14		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.13	✓	1404	1.1	even	1	trivial
243.2.i.a.61.13	yes	1404	243.61	even	81	inner
729.2.i.a.64.14		1404	3.2	odd	2
729.2.i.a.262.14		1404	243.182	odd	162