Embedded newform 243.2.i.a.4.7

• Label: 243.2.i.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55794 - 0.0604553i) q^{2} +(-0.167548 + 1.72393i) q^{3} +(0.429550 + 0.0333873i) q^{4} +(1.31451 + 1.62974i) q^{5} +(0.365251 - 2.67566i) q^{6} +(3.10528 - 1.41152i) q^{7} +(2.42995 + 0.284021i) q^{8} +(-2.94386 - 0.577680i) q^{9} +(-1.94941 - 2.61851i) q^{10} +(0.0304801 + 1.57154i) q^{11} +(-0.129527 + 0.734920i) q^{12} +(0.525951 + 0.168639i) q^{13} +(-4.92318 + 2.01134i) q^{14} +(-3.02979 + 1.99306i) q^{15} +(-4.61990 - 0.722540i) q^{16} +(1.80702 + 6.03586i) q^{17} +(4.55144 + 1.07797i) q^{18} +(-1.67765 + 0.397609i) q^{19} +(0.510236 + 0.743943i) q^{20} +(1.91308 + 5.58977i) q^{21} +(0.0475220 - 2.45022i) q^{22} +(-4.55831 + 3.25808i) q^{23} +(-0.896764 + 4.14148i) q^{24} +(0.130413 - 0.602052i) q^{25} +(-0.809207 - 0.294527i) q^{26} +(1.48912 - 4.97821i) q^{27} +(1.38100 - 0.502643i) q^{28} +(-0.244091 - 1.78707i) q^{29} +(4.84074 - 2.92192i) q^{30} +(4.85599 + 4.76273i) q^{31} +(2.35260 + 0.462036i) q^{32} +(-2.71433 - 0.210763i) q^{33} +(-2.45034 - 9.51278i) q^{34} +(6.38233 + 3.20533i) q^{35} +(-1.24525 - 0.346430i) q^{36} +(-2.42123 - 1.59246i) q^{37} +(2.63772 - 0.518031i) q^{38} +(-0.378844 + 0.878446i) q^{39} +(2.73132 + 4.33353i) q^{40} +(-3.89581 + 7.39601i) q^{41} +(-2.64254 - 8.82421i) q^{42} +(0.204131 - 0.596596i) q^{43} +(-0.0393769 + 0.676075i) q^{44} +(-2.92826 - 5.55708i) q^{45} +(7.29856 - 4.80034i) q^{46} +(7.16315 - 7.02557i) q^{47} +(2.01966 - 7.84332i) q^{48} +(3.04770 - 3.49228i) q^{49} +(-0.239573 + 0.930080i) q^{50} +(-10.7082 + 2.10388i) q^{51} +(0.220292 + 0.0899992i) q^{52} +(-7.52397 - 6.31336i) q^{53} +(-2.62092 + 7.66574i) q^{54} +(-2.52114 + 2.11549i) q^{55} +(7.94658 - 2.54797i) q^{56} +(-0.404364 - 2.95876i) q^{57} +(0.272243 + 2.79891i) q^{58} +(2.10585 - 1.16196i) q^{59} +(-1.36799 + 0.754965i) q^{60} +(6.46228 - 0.502288i) q^{61} +(-7.27744 - 7.71363i) q^{62} +(-9.95690 + 2.36146i) q^{63} +(5.46275 + 1.29470i) q^{64} +(0.416530 + 1.07884i) q^{65} +(4.21604 + 0.492453i) q^{66} +(-0.405147 + 2.96620i) q^{67} +(0.574685 + 2.65304i) q^{68} +(-4.85296 - 8.40408i) q^{69} +(-9.74954 - 5.37957i) q^{70} +(5.68556 - 13.1806i) q^{71} +(-6.98936 - 2.23985i) q^{72} +(-1.41300 + 1.89799i) q^{73} +(3.67586 + 2.62735i) q^{74} +(1.01604 + 0.325695i) q^{75} +(-0.733908 + 0.114781i) q^{76} +(2.31292 + 4.83706i) q^{77} +(0.643325 - 1.34567i) q^{78} +(8.32526 - 13.2089i) q^{79} +(-4.89536 - 8.47902i) q^{80} +(8.33257 + 3.40121i) q^{81} +(6.51659 - 11.2871i) q^{82} +(-3.63882 - 6.90813i) q^{83} +(0.635137 + 2.46496i) q^{84} +(-7.46153 + 10.8792i) q^{85} +(-0.354092 + 0.917122i) q^{86} +(3.12167 - 0.121377i) q^{87} +(-0.372286 + 3.82743i) q^{88} +(5.72803 + 13.2791i) q^{89} +(4.22612 + 8.83465i) q^{90} +(1.87126 - 0.218719i) q^{91} +(-2.06680 + 1.24732i) q^{92} +(-9.02421 + 7.57340i) q^{93} +(-11.5845 + 10.5124i) q^{94} +(-2.85328 - 2.21146i) q^{95} +(-1.19069 + 3.97830i) q^{96} +(-7.37002 + 9.13739i) q^{97} +(-4.95928 + 5.25653i) q^{98} +(0.818121 - 4.64401i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55794	−	0.0604553i	−1.10163	−	0.0427484i	−0.518464	−	0.855099i	\(-0.673496\pi\)
							−0.583169	+	0.812351i	\(0.698187\pi\)
\(3\)	−0.167548	+	1.72393i	−0.0967337	+	0.995310i
							\(5\)	1.31451	+	1.62974i	0.587867	+	0.728841i	0.981720	−	0.190333i	\(-0.0609567\pi\)
−0.393852	+	0.919174i	\(0.628858\pi\)
\(7\)	3.10528	−	1.41152i	1.17368	−	0.533505i	0.270484	−	0.962724i	\(-0.412816\pi\)
							0.903200	+	0.429219i	\(0.141211\pi\)
\(11\)	0.0304801	+	1.57154i	0.00919008	+	0.473838i	0.975931	+	0.218080i	\(0.0699794\pi\)
							−0.966741	+	0.255758i	\(0.917675\pi\)
\(13\)	0.525951	+	0.168639i	0.145873	+	0.0467721i	0.377365	−	0.926064i	\(-0.376830\pi\)
							−0.231493	+	0.972837i	\(0.574361\pi\)
\(17\)	1.80702	+	6.03586i	0.438266	+	1.46391i	0.837182	+	0.546924i	\(0.184201\pi\)
							−0.398916	+	0.916988i	\(0.630613\pi\)
\(19\)	−1.67765	+	0.397609i	−0.384878	+	0.0912178i	−0.418500	−	0.908217i	\(-0.637444\pi\)
							0.0336217	+	0.999435i	\(0.489296\pi\)
\(23\)	−4.55831	+	3.25808i	−0.950473	+	0.679357i	−0.947202	−	0.320638i	\(-0.896103\pi\)
							−0.00327068	+	0.999995i	\(0.501041\pi\)
\(29\)	−0.244091	−	1.78707i	−0.0453266	−	0.331850i	−0.999472	−	0.0325047i	\(-0.989652\pi\)
							0.954145	−	0.299345i	\(-0.0967681\pi\)
\(31\)	4.85599	+	4.76273i	0.872162	+	0.855411i	0.990493	−	0.137563i	\(-0.0439271\pi\)
							−0.118331	+	0.992974i	\(0.537754\pi\)
\(37\)	−2.42123	−	1.59246i	−0.398047	−	0.261800i	0.334667	−	0.942337i	\(-0.391376\pi\)
							−0.732714	+	0.680537i	\(0.761747\pi\)
\(41\)	−3.89581	+	7.39601i	−0.608423	+	1.15506i	0.366057	+	0.930592i	\(0.380707\pi\)
							−0.974481	+	0.224471i	\(0.927935\pi\)
\(43\)	0.204131	−	0.596596i	0.0311297	−	0.0909800i	−0.929489	−	0.368850i	\(-0.879752\pi\)
							0.960619	+	0.277870i	\(0.0896283\pi\)
\(47\)	7.16315	−	7.02557i	1.04485	−	1.02478i	0.0451862	−	0.998979i	\(-0.485612\pi\)
							0.999667	−	0.0258062i	\(-0.00821528\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.7	✓	1404
3.2	odd	2		729.2.i.a.64.20		1404
243.61	even	81	inner	243.2.i.a.61.7	yes	1404
243.182	odd	162		729.2.i.a.262.20		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.7	✓	1404	1.1	even	1	trivial
243.2.i.a.61.7	yes	1404	243.61	even	81	inner
729.2.i.a.64.20		1404	3.2	odd	2
729.2.i.a.262.20		1404	243.182	odd	162