Embedded newform 243.2.i.a.4.12

• Label: 243.2.i.a.4.12
• 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			4.12
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.479029 - 0.0185885i) q^{2} +(-1.73158 - 0.0404413i) q^{3} +(-1.76486 - 0.137176i) q^{4} +(-0.169194 - 0.209768i) q^{5} +(0.828725 + 0.0515600i) q^{6} +(0.242417 - 0.110192i) q^{7} +(1.79517 + 0.209825i) q^{8} +(2.99673 + 0.140055i) q^{9} +(0.0771496 + 0.103630i) q^{10} +(0.0282943 + 1.45884i) q^{11} +(3.05045 + 0.308904i) q^{12} +(1.74620 + 0.559897i) q^{13} +(-0.118173 + 0.0482791i) q^{14} +(0.284490 + 0.370072i) q^{15} +(2.64181 + 0.413172i) q^{16} +(1.76246 + 5.88703i) q^{17} +(-1.43292 - 0.122795i) q^{18} +(5.89204 - 1.39644i) q^{19} +(0.269829 + 0.393421i) q^{20} +(-0.424221 + 0.181003i) q^{21} +(0.0135640 - 0.699355i) q^{22} +(0.490512 - 0.350597i) q^{23} +(-3.09999 - 0.435927i) q^{24} +(1.04314 - 4.81569i) q^{25} +(-0.826073 - 0.300666i) q^{26} +(-5.18341 - 0.363707i) q^{27} +(-0.442949 + 0.161220i) q^{28} +(1.04708 + 7.66600i) q^{29} +(-0.129400 - 0.182563i) q^{30} +(-2.42178 - 2.37526i) q^{31} +(-4.80484 - 0.943640i) q^{32} +(0.0100039 - 2.52725i) q^{33} +(-0.734839 - 2.85282i) q^{34} +(-0.0641303 - 0.0322075i) q^{35} +(-5.26960 - 0.658257i) q^{36} +(3.67594 + 2.41770i) q^{37} +(-2.84841 + 0.559410i) q^{38} +(-3.00104 - 1.04012i) q^{39} +(-0.259717 - 0.412069i) q^{40} +(2.25060 - 4.27266i) q^{41} +(0.206579 - 0.0788199i) q^{42} +(1.20148 - 3.51146i) q^{43} +(0.150183 - 2.57854i) q^{44} +(-0.477650 - 0.652314i) q^{45} +(-0.241487 + 0.158828i) q^{46} +(-3.78017 + 3.70757i) q^{47} +(-4.55780 - 0.822279i) q^{48} +(-4.55603 + 5.22063i) q^{49} +(-0.589213 + 2.28746i) q^{50} +(-2.81376 - 10.2651i) q^{51} +(-3.00500 - 1.22768i) q^{52} +(6.69709 + 5.61953i) q^{53} +(2.47624 + 0.270578i) q^{54} +(0.301231 - 0.252763i) q^{55} +(0.458301 - 0.146948i) q^{56} +(-10.2590 + 2.17976i) q^{57} +(-0.359083 - 3.69170i) q^{58} +(-4.87739 + 2.69123i) q^{59} +(-0.451320 - 0.692151i) q^{60} +(15.3725 - 1.19485i) q^{61} +(1.11595 + 1.18284i) q^{62} +(0.741892 - 0.296264i) q^{63} +(-2.91959 - 0.691954i) q^{64} +(-0.177999 - 0.461028i) q^{65} +(-0.0517699 + 1.21044i) q^{66} +(-0.147970 + 1.08333i) q^{67} +(-2.30294 - 10.6316i) q^{68} +(-0.863539 + 0.587249i) q^{69} +(0.0301216 + 0.0166204i) q^{70} +(2.32719 - 5.39504i) q^{71} +(5.35024 + 0.880210i) q^{72} +(-2.28918 + 3.07491i) q^{73} +(-1.71594 - 1.22648i) q^{74} +(-2.00104 + 8.29655i) q^{75} +(-10.5902 + 1.65628i) q^{76} +(0.167612 + 0.350531i) q^{77} +(1.41825 + 0.554035i) q^{78} +(0.810888 - 1.28656i) q^{79} +(-0.360309 - 0.624074i) q^{80} +(8.96077 + 0.839412i) q^{81} +(-1.15753 + 2.00489i) q^{82} +(4.86840 + 9.24243i) q^{83} +(0.773521 - 0.261252i) q^{84} +(0.936711 - 1.36576i) q^{85} +(-0.640817 + 1.65976i) q^{86} +(-1.50308 - 13.3166i) q^{87} +(-0.255309 + 2.62481i) q^{88} +(-2.01691 - 4.67572i) q^{89} +(0.216683 + 0.321356i) q^{90} +(0.485006 - 0.0566890i) q^{91} +(-0.913780 + 0.551469i) q^{92} +(4.09744 + 4.21089i) q^{93} +(1.87973 - 1.70576i) q^{94} +(-1.28983 - 0.999690i) q^{95} +(8.28180 + 1.82830i) q^{96} +(-6.30856 + 7.82139i) q^{97} +(2.27951 - 2.41614i) q^{98} +(-0.119528 + 4.37572i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.479029	−	0.0185885i	−0.338725	−	0.0131441i	−0.131148	−	0.991363i	\(-0.541866\pi\)
							−0.207577	+	0.978219i	\(0.566558\pi\)
\(3\)	−1.73158	−	0.0404413i	−0.999727	−	0.0233488i
							\(5\)	−0.169194	−	0.209768i	−0.0756659	−	0.0938110i	0.739117	−	0.673577i	\(-0.235243\pi\)
−0.814783	+	0.579766i	\(0.803144\pi\)
\(7\)	0.242417	−	0.110192i	0.0916251	−	0.0416487i	−0.367474	−	0.930034i	\(-0.619777\pi\)
							0.459099	+	0.888385i	\(0.348172\pi\)
\(11\)	0.0282943	+	1.45884i	0.00853104	+	0.439858i	0.979589	+	0.201013i	\(0.0644232\pi\)
							−0.971058	+	0.238845i	\(0.923231\pi\)
\(13\)	1.74620	+	0.559897i	0.484309	+	0.155287i	0.537443	−	0.843300i	\(-0.319390\pi\)
							−0.0531342	+	0.998587i	\(0.516921\pi\)
\(17\)	1.76246	+	5.88703i	0.427459	+	1.42781i	0.852542	+	0.522659i	\(0.175060\pi\)
							−0.425082	+	0.905155i	\(0.639755\pi\)
\(19\)	5.89204	−	1.39644i	1.35173	−	0.320365i	0.509955	−	0.860201i	\(-0.329662\pi\)
							0.841770	+	0.539836i	\(0.181514\pi\)
\(23\)	0.490512	−	0.350597i	0.102279	−	0.0731045i	−0.529203	−	0.848495i	\(-0.677509\pi\)
							0.631482	+	0.775391i	\(0.282447\pi\)
\(29\)	1.04708	+	7.66600i	0.194438	+	1.42354i	0.786472	+	0.617626i	\(0.211906\pi\)
							−0.592033	+	0.805913i	\(0.701675\pi\)
\(31\)	−2.42178	−	2.37526i	−0.434964	−	0.426609i	0.449455	−	0.893303i	\(-0.351618\pi\)
							−0.884419	+	0.466693i	\(0.845445\pi\)
\(37\)	3.67594	+	2.41770i	0.604320	+	0.397468i	0.814473	−	0.580201i	\(-0.197026\pi\)
							−0.210153	+	0.977668i	\(0.567396\pi\)
\(41\)	2.25060	−	4.27266i	0.351485	−	0.667277i	−0.643702	−	0.765276i	\(-0.722602\pi\)
							0.995187	+	0.0979991i	\(0.0312442\pi\)
\(43\)	1.20148	−	3.51146i	0.183224	−	0.535493i	−0.815861	−	0.578247i	\(-0.803737\pi\)
							0.999086	+	0.0427547i	\(0.0136134\pi\)
\(47\)	−3.78017	+	3.70757i	−0.551395	+	0.540804i	−0.921598	−	0.388147i	\(-0.873115\pi\)
							0.370203	+	0.928951i	\(0.379288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.12	✓	1404
3.2	odd	2		729.2.i.a.64.15		1404
243.61	even	81	inner	243.2.i.a.61.12	yes	1404
243.182	odd	162		729.2.i.a.262.15		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.12	✓	1404	1.1	even	1	trivial
243.2.i.a.61.12	yes	1404	243.61	even	81	inner
729.2.i.a.64.15		1404	3.2	odd	2
729.2.i.a.262.15		1404	243.182	odd	162