Embedded newform 243.2.i.a.4.16

• Label: 243.2.i.a.4.16
• 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			4.16
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.498870 + 0.0193584i) q^{2} +(0.114249 - 1.72828i) q^{3} +(-1.74549 - 0.135670i) q^{4} +(0.793310 + 0.983550i) q^{5} +(0.0904521 - 0.859974i) q^{6} +(4.69382 - 2.13360i) q^{7} +(-1.85988 - 0.217389i) q^{8} +(-2.97389 - 0.394908i) q^{9} +(0.376718 + 0.506021i) q^{10} +(-0.0809506 - 4.17379i) q^{11} +(-0.433896 + 3.00119i) q^{12} +(-3.54298 - 1.13601i) q^{13} +(2.38291 - 0.973525i) q^{14} +(1.79048 - 1.25869i) q^{15} +(2.53582 + 0.396595i) q^{16} +(0.721966 + 2.41153i) q^{17} +(-1.47594 - 0.254578i) q^{18} +(1.53658 - 0.364176i) q^{19} +(-1.25128 - 1.82440i) q^{20} +(-3.15119 - 8.35599i) q^{21} +(0.0404142 - 2.08375i) q^{22} +(0.506139 - 0.361766i) q^{23} +(-0.588199 + 3.18956i) q^{24} +(0.720490 - 3.32615i) q^{25} +(-1.74550 - 0.635309i) q^{26} +(-1.02228 + 5.09460i) q^{27} +(-8.48248 + 3.08737i) q^{28} +(1.02541 + 7.50731i) q^{29} +(0.917584 - 0.593262i) q^{30} +(4.69293 + 4.60279i) q^{31} +(4.93226 + 0.968664i) q^{32} +(-7.22272 - 0.336946i) q^{33} +(0.313484 + 1.21702i) q^{34} +(5.82216 + 2.92400i) q^{35} +(5.13732 + 1.09278i) q^{36} +(0.272573 + 0.179274i) q^{37} +(0.773603 - 0.151931i) q^{38} +(-2.36813 + 5.99348i) q^{39} +(-1.26165 - 2.00175i) q^{40} +(-5.77739 + 10.9681i) q^{41} +(-1.41028 - 4.22955i) q^{42} +(0.601044 - 1.75662i) q^{43} +(-0.424961 + 7.29629i) q^{44} +(-1.97081 - 3.23826i) q^{45} +(0.259500 - 0.170676i) q^{46} +(-4.91401 + 4.81962i) q^{47} +(0.975143 - 4.33730i) q^{48} +(12.8770 - 14.7554i) q^{49} +(0.423820 - 1.64537i) q^{50} +(4.25029 - 0.972243i) q^{51} +(6.03012 + 2.46357i) q^{52} +(4.44061 + 3.72611i) q^{53} +(-0.608606 + 2.52175i) q^{54} +(4.04091 - 3.39073i) q^{55} +(-9.19378 + 2.94787i) q^{56} +(-0.453845 - 2.69725i) q^{57} +(0.366215 + 3.76502i) q^{58} +(9.00067 - 4.96636i) q^{59} +(-3.29604 + 1.95412i) q^{60} +(-2.04933 + 0.159287i) q^{61} +(2.25206 + 2.38704i) q^{62} +(-14.8015 + 4.49148i) q^{63} +(-2.55312 - 0.605101i) q^{64} +(-1.69336 - 4.38591i) q^{65} +(-3.59668 - 0.307913i) q^{66} +(0.617949 - 4.52419i) q^{67} +(-0.933011 - 4.30726i) q^{68} +(-0.567407 - 0.916080i) q^{69} +(2.84789 + 1.57140i) q^{70} +(-4.61055 + 10.6885i) q^{71} +(5.44525 + 1.38098i) q^{72} +(-0.380486 + 0.511081i) q^{73} +(0.132508 + 0.0947111i) q^{74} +(-5.66620 - 1.62522i) q^{75} +(-2.73149 + 0.427198i) q^{76} +(-9.28518 - 19.4183i) q^{77} +(-1.29741 + 2.94412i) q^{78} +(-5.64359 + 8.95417i) q^{79} +(1.62162 + 2.80873i) q^{80} +(8.68809 + 2.34883i) q^{81} +(-3.09449 + 5.35982i) q^{82} +(-6.40500 - 12.1596i) q^{83} +(4.36672 + 15.0128i) q^{84} +(-1.79912 + 2.62318i) q^{85} +(0.333848 - 0.864689i) q^{86} +(13.0919 - 0.914487i) q^{87} +(-0.756779 + 7.78037i) q^{88} +(1.76038 + 4.08101i) q^{89} +(-0.920489 - 1.65362i) q^{90} +(-19.0539 + 2.22708i) q^{91} +(-0.932540 + 0.562791i) q^{92} +(8.49108 - 7.58483i) q^{93} +(-2.54475 + 2.30924i) q^{94} +(1.57717 + 1.22240i) q^{95} +(2.23763 - 8.41365i) q^{96} +(-0.103363 + 0.128149i) q^{97} +(6.70960 - 7.11176i) q^{98} +(-1.40753 + 12.4444i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.498870	+	0.0193584i	0.352754	+	0.0136885i	0.214544	−	0.976714i	\(-0.431174\pi\)
							0.138211	+	0.990403i	\(0.455865\pi\)
\(3\)	0.114249	−	1.72828i	0.0659617	−	0.997822i
							\(5\)	0.793310	+	0.983550i	0.354779	+	0.439857i	0.924038	−	0.382302i	\(-0.124868\pi\)
−0.569259	+	0.822158i	\(0.692770\pi\)
\(7\)	4.69382	−	2.13360i	1.77410	−	0.806426i	0.793996	−	0.607923i	\(-0.207997\pi\)
							0.980100	−	0.198503i	\(-0.0636080\pi\)
\(11\)	−0.0809506	−	4.17379i	−0.0244075	−	1.25845i	−0.789187	−	0.614153i	\(-0.789498\pi\)
							0.764779	−	0.644292i	\(-0.222848\pi\)
\(13\)	−3.54298	−	1.13601i	−0.982647	−	0.315073i	−0.229778	−	0.973243i	\(-0.573800\pi\)
							−0.752869	+	0.658170i	\(0.771331\pi\)
\(17\)	0.721966	+	2.41153i	0.175103	+	0.584883i	0.999813	+	0.0193283i	\(0.00615276\pi\)
							−0.824711	+	0.565555i	\(0.808662\pi\)
\(19\)	1.53658	−	0.364176i	0.352516	−	0.0835477i	−0.0505430	−	0.998722i	\(-0.516095\pi\)
							0.403059	+	0.915174i	\(0.367947\pi\)
\(23\)	0.506139	−	0.361766i	0.105537	−	0.0754334i	−0.527499	−	0.849556i	\(-0.676870\pi\)
							0.633036	+	0.774122i	\(0.281808\pi\)
\(29\)	1.02541	+	7.50731i	0.190413	+	1.39407i	0.799356	+	0.600858i	\(0.205174\pi\)
							−0.608942	+	0.793215i	\(0.708406\pi\)
\(31\)	4.69293	+	4.60279i	0.842875	+	0.826686i	0.986638	−	0.162926i	\(-0.0520931\pi\)
							−0.143763	+	0.989612i	\(0.545920\pi\)
\(37\)	0.272573	+	0.179274i	0.0448108	+	0.0294725i	0.571717	−	0.820451i	\(-0.306278\pi\)
							−0.526906	+	0.849924i	\(0.676648\pi\)
\(41\)	−5.77739	+	10.9681i	−0.902277	+	1.71293i	−0.229168	+	0.973387i	\(0.573601\pi\)
							−0.673109	+	0.739544i	\(0.735041\pi\)
\(43\)	0.601044	−	1.75662i	0.0916584	−	0.267882i	−0.890790	−	0.454415i	\(-0.849849\pi\)
							0.982449	+	0.186533i	\(0.0597251\pi\)
\(47\)	−4.91401	+	4.81962i	−0.716781	+	0.703014i	−0.963987	−	0.265949i	\(-0.914315\pi\)
							0.247206	+	0.968963i	\(0.420488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.16	✓	1404
3.2	odd	2		729.2.i.a.64.11		1404
243.61	even	81	inner	243.2.i.a.61.16	yes	1404
243.182	odd	162		729.2.i.a.262.11		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.16	✓	1404	1.1	even	1	trivial
243.2.i.a.61.16	yes	1404	243.61	even	81	inner
729.2.i.a.64.11		1404	3.2	odd	2
729.2.i.a.262.11		1404	243.182	odd	162