Embedded newform 243.2.i.a.4.2

• Label: 243.2.i.a.4.2
• 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			4.2
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.42984 - 0.0942887i) q^{2} +(1.64930 - 0.528965i) q^{3} +(3.90123 + 0.303227i) q^{4} +(-0.723370 - 0.896838i) q^{5} +(-4.05741 + 1.12979i) q^{6} +(1.78613 - 0.811895i) q^{7} +(-4.62031 - 0.540037i) q^{8} +(2.44039 - 1.74485i) q^{9} +(1.67311 + 2.24737i) q^{10} +(-0.0219513 - 1.13180i) q^{11} +(6.59470 - 1.56350i) q^{12} +(-1.15303 - 0.369705i) q^{13} +(-4.41655 + 1.80436i) q^{14} +(-1.66745 - 1.09652i) q^{15} +(3.44368 + 0.538581i) q^{16} +(1.18796 + 3.96805i) q^{17} +(-6.09427 + 4.00959i) q^{18} +(-1.74384 + 0.413297i) q^{19} +(-2.55008 - 3.71811i) q^{20} +(2.51640 - 2.28386i) q^{21} +(-0.0533782 + 2.75216i) q^{22} +(4.47543 - 3.19884i) q^{23} +(-7.90594 + 1.55330i) q^{24} +(0.777465 - 3.58918i) q^{25} +(2.76682 + 1.00704i) q^{26} +(3.10198 - 4.16866i) q^{27} +(7.21429 - 2.62579i) q^{28} +(-1.40567 - 10.2913i) q^{29} +(3.94824 + 2.82158i) q^{30} +(2.37780 + 2.33213i) q^{31} +(0.812347 + 0.159540i) q^{32} +(-0.634888 - 1.85507i) q^{33} +(-2.51240 - 9.75373i) q^{34} +(-2.02017 - 1.01457i) q^{35} +(10.0496 - 6.06705i) q^{36} +(0.649096 + 0.426917i) q^{37} +(4.27621 - 0.839821i) q^{38} +(-2.09726 + 0.000158814i) q^{39} +(2.85787 + 4.53431i) q^{40} +(-4.25842 + 8.08441i) q^{41} +(-6.32979 + 5.31214i) q^{42} +(2.07626 - 6.06810i) q^{43} +(0.257557 - 4.42208i) q^{44} +(-3.33015 - 0.926467i) q^{45} +(-11.1762 + 7.35068i) q^{46} +(-2.13280 + 2.09184i) q^{47} +(5.96455 - 0.933301i) q^{48} +(-2.07157 + 2.37375i) q^{49} +(-2.22753 + 8.64781i) q^{50} +(4.05826 + 5.91613i) q^{51} +(-4.38614 - 1.79193i) q^{52} +(10.6804 + 8.96189i) q^{53} +(-7.93036 + 9.83668i) q^{54} +(-0.999164 + 0.838398i) q^{55} +(-8.69092 + 2.78663i) q^{56} +(-2.65750 + 1.60408i) q^{57} +(2.44519 + 25.1387i) q^{58} +(-5.10048 + 2.81432i) q^{59} +(-6.17261 - 4.78339i) q^{60} +(-0.538800 + 0.0418789i) q^{61} +(-5.55778 - 5.89090i) q^{62} +(2.94222 - 5.09786i) q^{63} +(-8.74200 - 2.07189i) q^{64} +(0.502503 + 1.30152i) q^{65} +(1.36776 + 4.56738i) q^{66} +(0.504746 - 3.69540i) q^{67} +(3.43127 + 15.8405i) q^{68} +(5.68925 - 7.64320i) q^{69} +(4.81302 + 2.65571i) q^{70} +(2.22269 - 5.15278i) q^{71} +(-12.2176 + 6.74382i) q^{72} +(-9.40286 + 12.6302i) q^{73} +(-1.53694 - 1.09854i) q^{74} +(-0.616275 - 6.33089i) q^{75} +(-6.92844 + 1.08359i) q^{76} +(-0.958113 - 2.00372i) q^{77} +(5.09601 + 0.197362i) q^{78} +(-9.33911 + 14.8175i) q^{79} +(-2.00803 - 3.47801i) q^{80} +(2.91103 - 8.51622i) q^{81} +(11.1095 - 19.2423i) q^{82} +(5.71501 + 10.8497i) q^{83} +(10.5096 - 8.14682i) q^{84} +(2.69937 - 3.93577i) q^{85} +(-5.61712 + 14.5487i) q^{86} +(-7.76211 - 16.2299i) q^{87} +(-0.509793 + 5.24113i) q^{88} +(-0.520452 - 1.20654i) q^{89} +(8.00436 + 2.56516i) q^{90} +(-2.35963 + 0.275801i) q^{91} +(18.4296 - 11.1223i) q^{92} +(5.15533 + 2.58862i) q^{93} +(5.37959 - 4.88172i) q^{94} +(1.63210 + 1.26497i) q^{95} +(1.42420 - 0.166574i) q^{96} +(-2.94313 + 3.64891i) q^{97} +(5.25739 - 5.57250i) q^{98} +(-2.02839 - 2.72374i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.42984	−	0.0942887i	−1.71815	−	0.0666722i	−0.839272	−	0.543712i	\(-0.817018\pi\)
							−0.878882	+	0.477040i	\(0.841710\pi\)
\(3\)	1.64930	−	0.528965i	0.952225	−	0.305398i
							\(5\)	−0.723370	−	0.896838i	−0.323501	−	0.401078i	0.590337	−	0.807157i	\(-0.298995\pi\)
−0.913838	+	0.406079i	\(0.866896\pi\)
\(7\)	1.78613	−	0.811895i	0.675093	−	0.306868i	−0.0467675	−	0.998906i	\(-0.514892\pi\)
							0.721861	+	0.692038i	\(0.243287\pi\)
\(11\)	−0.0219513	−	1.13180i	−0.00661856	−	0.341251i	−0.988454	−	0.151519i	\(-0.951584\pi\)
							0.981836	−	0.189732i	\(-0.0607621\pi\)
\(13\)	−1.15303	−	0.369705i	−0.319794	−	0.102538i	0.141094	−	0.989996i	\(-0.454938\pi\)
							−0.460888	+	0.887458i	\(0.652469\pi\)
\(17\)	1.18796	+	3.96805i	0.288122	+	0.962394i	0.972429	+	0.233200i	\(0.0749197\pi\)
							−0.684307	+	0.729194i	\(0.739895\pi\)
\(19\)	−1.74384	+	0.413297i	−0.400064	+	0.0948169i	−0.425722	−	0.904854i	\(-0.639980\pi\)
							0.0256583	+	0.999671i	\(0.491832\pi\)
\(23\)	4.47543	−	3.19884i	0.933191	−	0.667005i	−0.00975432	−	0.999952i	\(-0.503105\pi\)
							0.942945	+	0.332948i	\(0.108043\pi\)
\(29\)	−1.40567	−	10.2913i	−0.261026	−	1.91105i	−0.392544	−	0.919733i	\(-0.628405\pi\)
							0.131518	−	0.991314i	\(-0.458015\pi\)
\(31\)	2.37780	+	2.33213i	0.427066	+	0.418863i	0.881697	−	0.471817i	\(-0.156402\pi\)
							−0.454631	+	0.890680i	\(0.650229\pi\)
\(37\)	0.649096	+	0.426917i	0.106711	+	0.0701848i	0.601743	−	0.798690i	\(-0.294473\pi\)
							−0.495032	+	0.868875i	\(0.664844\pi\)
\(41\)	−4.25842	+	8.08441i	−0.665054	+	1.26257i	0.287377	+	0.957818i	\(0.407217\pi\)
							−0.952431	+	0.304756i	\(0.901425\pi\)
\(43\)	2.07626	−	6.06810i	0.316627	−	0.925376i	−0.666984	−	0.745072i	\(-0.732415\pi\)
							0.983611	−	0.180304i	\(-0.0577083\pi\)
\(47\)	−2.13280	+	2.09184i	−0.311101	+	0.305126i	−0.838946	−	0.544215i	\(-0.816827\pi\)
							0.527845	+	0.849341i	\(0.323000\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.2	✓	1404
3.2	odd	2		729.2.i.a.64.25		1404
243.61	even	81	inner	243.2.i.a.61.2	yes	1404
243.182	odd	162		729.2.i.a.262.25		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.2	✓	1404	1.1	even	1	trivial
243.2.i.a.61.2	yes	1404	243.61	even	81	inner
729.2.i.a.64.25		1404	3.2	odd	2
729.2.i.a.262.25		1404	243.182	odd	162