Embedded newform 243.2.i.a.4.19

• Label: 243.2.i.a.4.19
• 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			4.19
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14737 + 0.0445232i) q^{2} +(-1.35679 - 1.07662i) q^{3} +(-0.679512 - 0.0528158i) q^{4} +(-1.27759 - 1.58397i) q^{5} +(-1.50880 - 1.29569i) q^{6} +(-1.57880 + 0.717652i) q^{7} +(-3.05824 - 0.357457i) q^{8} +(0.681758 + 2.92151i) q^{9} +(-1.39535 - 1.87428i) q^{10} +(0.00295836 + 0.152532i) q^{11} +(0.865092 + 0.803239i) q^{12} +(-3.13701 - 1.00584i) q^{13} +(-1.84342 + 0.753119i) q^{14} +(0.0280874 + 3.52460i) q^{15} +(-2.14626 - 0.335669i) q^{16} +(-1.86084 - 6.21564i) q^{17} +(0.652154 + 3.38240i) q^{18} +(2.11967 - 0.502371i) q^{19} +(0.784482 + 1.14380i) q^{20} +(2.91474 + 0.726070i) q^{21} +(-0.00339688 + 0.175142i) q^{22} +(4.41218 - 3.15364i) q^{23} +(3.76454 + 3.77757i) q^{24} +(0.181811 - 0.839331i) q^{25} +(-3.55453 - 1.29374i) q^{26} +(2.22037 - 4.69787i) q^{27} +(1.11072 - 0.404267i) q^{28} +(0.652513 + 4.77724i) q^{29} +(-0.124700 + 4.04527i) q^{30} +(-1.14467 - 1.12269i) q^{31} +(3.59507 + 0.706049i) q^{32} +(0.160206 - 0.210139i) q^{33} +(-1.85833 - 7.21449i) q^{34} +(3.15380 + 1.58390i) q^{35} +(-0.308961 - 2.02121i) q^{36} +(-5.69930 - 3.74849i) q^{37} +(2.45441 - 0.482031i) q^{38} +(3.17335 + 4.74210i) q^{39} +(3.34099 + 5.30084i) q^{40} +(-0.728644 + 1.38330i) q^{41} +(3.31196 + 0.962844i) q^{42} +(-4.01097 + 11.7225i) q^{43} +(0.00604587 - 0.103804i) q^{44} +(3.75657 - 4.81238i) q^{45} +(5.20281 - 3.42194i) q^{46} +(7.45766 - 7.31442i) q^{47} +(2.55064 + 2.76615i) q^{48} +(-2.62507 + 3.00800i) q^{49} +(0.245974 - 0.954928i) q^{50} +(-4.16715 + 10.4368i) q^{51} +(2.07851 + 0.849166i) q^{52} +(-3.65378 - 3.06589i) q^{53} +(2.75674 - 5.29133i) q^{54} +(0.237826 - 0.199560i) q^{55} +(5.08487 - 1.63040i) q^{56} +(-3.41681 - 1.60048i) q^{57} +(0.535976 + 5.51031i) q^{58} +(-0.0820422 + 0.0452690i) q^{59} +(0.167069 - 2.39649i) q^{60} +(-12.6243 + 0.981241i) q^{61} +(-1.26338 - 1.33910i) q^{62} +(-3.17298 - 4.12321i) q^{63} +(8.32103 + 1.97212i) q^{64} +(2.41461 + 6.25399i) q^{65} +(0.193171 - 0.233974i) q^{66} +(0.745922 - 5.46111i) q^{67} +(0.936179 + 4.32189i) q^{68} +(-9.38169 - 0.471440i) q^{69} +(3.54805 + 1.95773i) q^{70} +(6.49231 - 15.0509i) q^{71} +(-1.04067 - 9.17836i) q^{72} +(1.07962 - 1.45018i) q^{73} +(-6.37231 - 4.55465i) q^{74} +(-1.15032 + 0.943054i) q^{75} +(-1.46687 + 0.229415i) q^{76} +(-0.114136 - 0.238694i) q^{77} +(3.42987 + 5.58223i) q^{78} +(-4.78233 + 7.58768i) q^{79} +(2.21036 + 3.82846i) q^{80} +(-8.07041 + 3.98352i) q^{81} +(-0.897612 + 1.55471i) q^{82} +(1.86354 + 3.53784i) q^{83} +(-1.94225 - 0.647318i) q^{84} +(-7.46798 + 10.8886i) q^{85} +(-5.12398 + 13.2714i) q^{86} +(4.25797 - 7.18423i) q^{87} +(0.0454763 - 0.467537i) q^{88} +(0.235678 + 0.546362i) q^{89} +(4.52443 - 5.35433i) q^{90} +(5.67455 - 0.663260i) q^{91} +(-3.16469 + 1.90990i) q^{92} +(0.344368 + 2.75564i) q^{93} +(8.88235 - 8.06030i) q^{94} +(-3.50382 - 2.71566i) q^{95} +(-4.11761 - 4.82850i) q^{96} +(4.75272 - 5.89245i) q^{97} +(-3.14585 + 3.33441i) q^{98} +(-0.443607 + 0.112633i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14737	+	0.0445232i	0.811313	+	0.0314826i	0.441093	−	0.897462i	\(-0.354591\pi\)
							0.370220	+	0.928944i	\(0.379282\pi\)
\(3\)	−1.35679	−	1.07662i	−0.783343	−	0.621590i
							\(5\)	−1.27759	−	1.58397i	−0.571358	−	0.708372i	0.407433	−	0.913235i	\(-0.366424\pi\)
−0.978791	+	0.204863i	\(0.934325\pi\)
\(7\)	−1.57880	+	0.717652i	−0.596730	+	0.271247i	−0.689320	−	0.724457i	\(-0.742090\pi\)
							0.0925899	+	0.995704i	\(0.470485\pi\)
\(11\)	0.00295836	+	0.152532i	0.000891978	+	0.0459902i	0.999993	+	0.00360888i	\(0.00114874\pi\)
							−0.999102	+	0.0423813i	\(0.986506\pi\)
\(13\)	−3.13701	−	1.00584i	−0.870051	−	0.278970i	−0.163423	−	0.986556i	\(-0.552254\pi\)
							−0.706628	+	0.707586i	\(0.749784\pi\)
\(17\)	−1.86084	−	6.21564i	−0.451320	−	1.50752i	−0.816924	−	0.576745i	\(-0.804323\pi\)
							0.365604	−	0.930770i	\(-0.380862\pi\)
\(19\)	2.11967	−	0.502371i	0.486286	−	0.115252i	0.0198427	−	0.999803i	\(-0.493683\pi\)
							0.466443	+	0.884551i	\(0.345535\pi\)
\(23\)	4.41218	−	3.15364i	0.920003	−	0.657579i	−0.0196132	−	0.999808i	\(-0.506243\pi\)
							0.939617	+	0.342229i	\(0.111182\pi\)
\(29\)	0.652513	+	4.77724i	0.121169	+	0.887112i	0.946555	+	0.322543i	\(0.104538\pi\)
							−0.825386	+	0.564569i	\(0.809043\pi\)
\(31\)	−1.14467	−	1.12269i	−0.205589	−	0.201641i	0.590126	−	0.807311i	\(-0.299078\pi\)
							−0.795716	+	0.605670i	\(0.792905\pi\)
\(37\)	−5.69930	−	3.74849i	−0.936959	−	0.616248i	−0.0134854	−	0.999909i	\(-0.504293\pi\)
							−0.923474	+	0.383661i	\(0.874663\pi\)
\(41\)	−0.728644	+	1.38330i	−0.113795	+	0.216034i	−0.935041	−	0.354540i	\(-0.884637\pi\)
							0.821246	+	0.570575i	\(0.193279\pi\)
\(43\)	−4.01097	+	11.7225i	−0.611667	+	1.78766i	0.00442619	+	0.999990i	\(0.498591\pi\)
							−0.616093	+	0.787673i	\(0.711285\pi\)
\(47\)	7.45766	−	7.31442i	1.08781	−	1.06692i	0.0903355	−	0.995911i	\(-0.471206\pi\)
							0.997476	−	0.0710065i	\(-0.0226211\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.19	✓	1404
3.2	odd	2		729.2.i.a.64.8		1404
243.61	even	81	inner	243.2.i.a.61.19	yes	1404
243.182	odd	162		729.2.i.a.262.8		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.19	✓	1404	1.1	even	1	trivial
243.2.i.a.61.19	yes	1404	243.61	even	81	inner
729.2.i.a.64.8		1404	3.2	odd	2
729.2.i.a.262.8		1404	243.182	odd	162