LMFDB - Group of Dirichlet characters of modulus 243

Show commands: Magma / Pari/GP / SageMath

comment:Define the Dirichlet group

sage:G = DirichletGroup(243)

gp:g = idealstar(,243,2)

magma:G = FullDirichletGroup(243);

Character group

Order	=	162	comment:Order sage:G.order() gp:g.no magma:Order(G);
Structure	=	$C_{162}$	comment:Group structure sage:sorted(g.order() for g in G.gens()) gp:g.cyc magma:PrimaryInvariants(G);
Generators	=	$\chi_{243}(2,\cdot)$	comment:Generators sage:G.gens() gp:g.gen magma:Generators(G);

First 32 of 162 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character	Orbit	Order	Primitive	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$11$	$13$	$14$	$16$
$\chi_{243}(1,\cdot)$	243.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{243}(2,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{1}{162}\right)$	$e\left(\frac{1}{81}\right)$	$e\left(\frac{23}{162}\right)$	$e\left(\frac{35}{81}\right)$	$e\left(\frac{1}{54}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{121}{162}\right)$	$e\left(\frac{4}{81}\right)$	$e\left(\frac{71}{162}\right)$	$e\left(\frac{2}{81}\right)$
$\chi_{243}(4,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{1}{81}\right)$	$e\left(\frac{2}{81}\right)$	$e\left(\frac{23}{81}\right)$	$e\left(\frac{70}{81}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{40}{81}\right)$	$e\left(\frac{8}{81}\right)$	$e\left(\frac{71}{81}\right)$	$e\left(\frac{4}{81}\right)$
$\chi_{243}(5,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{23}{162}\right)$	$e\left(\frac{23}{81}\right)$	$e\left(\frac{43}{162}\right)$	$e\left(\frac{76}{81}\right)$	$e\left(\frac{23}{54}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{29}{162}\right)$	$e\left(\frac{11}{81}\right)$	$e\left(\frac{13}{162}\right)$	$e\left(\frac{46}{81}\right)$
$\chi_{243}(7,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{35}{81}\right)$	$e\left(\frac{70}{81}\right)$	$e\left(\frac{76}{81}\right)$	$e\left(\frac{20}{81}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{23}{81}\right)$	$e\left(\frac{37}{81}\right)$	$e\left(\frac{55}{81}\right)$	$e\left(\frac{59}{81}\right)$
$\chi_{243}(8,\cdot)$	243.h	54	no	$-1$	$1$	$e\left(\frac{1}{54}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{23}{54}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{17}{54}\right)$	$e\left(\frac{2}{27}\right)$
$\chi_{243}(10,\cdot)$	243.g	27	no	$1$	$1$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{16}{27}\right)$
$\chi_{243}(11,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{121}{162}\right)$	$e\left(\frac{40}{81}\right)$	$e\left(\frac{29}{162}\right)$	$e\left(\frac{23}{81}\right)$	$e\left(\frac{13}{54}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{61}{162}\right)$	$e\left(\frac{79}{81}\right)$	$e\left(\frac{5}{162}\right)$	$e\left(\frac{80}{81}\right)$
$\chi_{243}(13,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{4}{81}\right)$	$e\left(\frac{8}{81}\right)$	$e\left(\frac{11}{81}\right)$	$e\left(\frac{37}{81}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{79}{81}\right)$	$e\left(\frac{32}{81}\right)$	$e\left(\frac{41}{81}\right)$	$e\left(\frac{16}{81}\right)$
$\chi_{243}(14,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{71}{162}\right)$	$e\left(\frac{71}{81}\right)$	$e\left(\frac{13}{162}\right)$	$e\left(\frac{55}{81}\right)$	$e\left(\frac{17}{54}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{5}{162}\right)$	$e\left(\frac{41}{81}\right)$	$e\left(\frac{19}{162}\right)$	$e\left(\frac{61}{81}\right)$
$\chi_{243}(16,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{2}{81}\right)$	$e\left(\frac{4}{81}\right)$	$e\left(\frac{46}{81}\right)$	$e\left(\frac{59}{81}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{80}{81}\right)$	$e\left(\frac{16}{81}\right)$	$e\left(\frac{61}{81}\right)$	$e\left(\frac{8}{81}\right)$
$\chi_{243}(17,\cdot)$	243.h	54	no	$-1$	$1$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{35}{54}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{22}{27}\right)$
$\chi_{243}(19,\cdot)$	243.g	27	no	$1$	$1$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{23}{27}\right)$
$\chi_{243}(20,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{25}{162}\right)$	$e\left(\frac{25}{81}\right)$	$e\left(\frac{89}{162}\right)$	$e\left(\frac{65}{81}\right)$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{109}{162}\right)$	$e\left(\frac{19}{81}\right)$	$e\left(\frac{155}{162}\right)$	$e\left(\frac{50}{81}\right)$
$\chi_{243}(22,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{61}{81}\right)$	$e\left(\frac{41}{81}\right)$	$e\left(\frac{26}{81}\right)$	$e\left(\frac{58}{81}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{10}{81}\right)$	$e\left(\frac{2}{81}\right)$	$e\left(\frac{38}{81}\right)$	$e\left(\frac{1}{81}\right)$
$\chi_{243}(23,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{65}{162}\right)$	$e\left(\frac{65}{81}\right)$	$e\left(\frac{37}{162}\right)$	$e\left(\frac{7}{81}\right)$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{89}{162}\right)$	$e\left(\frac{17}{81}\right)$	$e\left(\frac{79}{162}\right)$	$e\left(\frac{49}{81}\right)$
$\chi_{243}(25,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{23}{81}\right)$	$e\left(\frac{46}{81}\right)$	$e\left(\frac{43}{81}\right)$	$e\left(\frac{71}{81}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{29}{81}\right)$	$e\left(\frac{22}{81}\right)$	$e\left(\frac{13}{81}\right)$	$e\left(\frac{11}{81}\right)$
$\chi_{243}(26,\cdot)$	243.f	18	no	$-1$	$1$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{9}\right)$
$\chi_{243}(28,\cdot)$	243.e	9	no	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{243}(29,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{37}{162}\right)$	$e\left(\frac{37}{81}\right)$	$e\left(\frac{41}{162}\right)$	$e\left(\frac{80}{81}\right)$	$e\left(\frac{37}{54}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{103}{162}\right)$	$e\left(\frac{67}{81}\right)$	$e\left(\frac{35}{162}\right)$	$e\left(\frac{74}{81}\right)$
$\chi_{243}(31,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{10}{81}\right)$	$e\left(\frac{20}{81}\right)$	$e\left(\frac{68}{81}\right)$	$e\left(\frac{52}{81}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{26}{27}\right)$	$e\left(\frac{76}{81}\right)$	$e\left(\frac{80}{81}\right)$	$e\left(\frac{62}{81}\right)$	$e\left(\frac{40}{81}\right)$
$\chi_{243}(32,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{5}{162}\right)$	$e\left(\frac{5}{81}\right)$	$e\left(\frac{115}{162}\right)$	$e\left(\frac{13}{81}\right)$	$e\left(\frac{5}{54}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{119}{162}\right)$	$e\left(\frac{20}{81}\right)$	$e\left(\frac{31}{162}\right)$	$e\left(\frac{10}{81}\right)$
$\chi_{243}(34,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{17}{81}\right)$	$e\left(\frac{34}{81}\right)$	$e\left(\frac{67}{81}\right)$	$e\left(\frac{56}{81}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{32}{81}\right)$	$e\left(\frac{55}{81}\right)$	$e\left(\frac{73}{81}\right)$	$e\left(\frac{68}{81}\right)$
$\chi_{243}(35,\cdot)$	243.h	54	no	$-1$	$1$	$e\left(\frac{31}{54}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{11}{54}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{8}{27}\right)$
$\chi_{243}(37,\cdot)$	243.g	27	no	$1$	$1$	$e\left(\frac{16}{27}\right)$	$e\left(\frac{5}{27}\right)$	$e\left(\frac{17}{27}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{19}{27}\right)$	$e\left(\frac{20}{27}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{10}{27}\right)$
$\chi_{243}(38,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{157}{162}\right)$	$e\left(\frac{76}{81}\right)$	$e\left(\frac{47}{162}\right)$	$e\left(\frac{68}{81}\right)$	$e\left(\frac{49}{54}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{43}{162}\right)$	$e\left(\frac{61}{81}\right)$	$e\left(\frac{131}{162}\right)$	$e\left(\frac{71}{81}\right)$
$\chi_{243}(40,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{13}{81}\right)$	$e\left(\frac{26}{81}\right)$	$e\left(\frac{56}{81}\right)$	$e\left(\frac{19}{81}\right)$	$e\left(\frac{13}{27}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{34}{81}\right)$	$e\left(\frac{23}{81}\right)$	$e\left(\frac{32}{81}\right)$	$e\left(\frac{52}{81}\right)$
$\chi_{243}(41,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{53}{162}\right)$	$e\left(\frac{53}{81}\right)$	$e\left(\frac{85}{162}\right)$	$e\left(\frac{73}{81}\right)$	$e\left(\frac{53}{54}\right)$	$e\left(\frac{23}{27}\right)$	$e\left(\frac{95}{162}\right)$	$e\left(\frac{50}{81}\right)$	$e\left(\frac{37}{162}\right)$	$e\left(\frac{25}{81}\right)$
$\chi_{243}(43,\cdot)$	243.i	81	yes	$1$	$1$	$e\left(\frac{65}{81}\right)$	$e\left(\frac{49}{81}\right)$	$e\left(\frac{37}{81}\right)$	$e\left(\frac{14}{81}\right)$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{8}{81}\right)$	$e\left(\frac{34}{81}\right)$	$e\left(\frac{79}{81}\right)$	$e\left(\frac{17}{81}\right)$
$\chi_{243}(44,\cdot)$	243.h	54	no	$-1$	$1$	$e\left(\frac{41}{54}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{25}{54}\right)$	$e\left(\frac{4}{27}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{47}{54}\right)$	$e\left(\frac{2}{27}\right)$	$e\left(\frac{49}{54}\right)$	$e\left(\frac{1}{27}\right)$
$\chi_{243}(46,\cdot)$	243.g	27	no	$1$	$1$	$e\left(\frac{11}{27}\right)$	$e\left(\frac{22}{27}\right)$	$e\left(\frac{10}{27}\right)$	$e\left(\frac{14}{27}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{8}{27}\right)$	$e\left(\frac{7}{27}\right)$	$e\left(\frac{25}{27}\right)$	$e\left(\frac{17}{27}\right)$
$\chi_{243}(47,\cdot)$	243.j	162	yes	$-1$	$1$	$e\left(\frac{61}{162}\right)$	$e\left(\frac{61}{81}\right)$	$e\left(\frac{107}{162}\right)$	$e\left(\frac{29}{81}\right)$	$e\left(\frac{7}{54}\right)$	$e\left(\frac{1}{27}\right)$	$e\left(\frac{91}{162}\right)$	$e\left(\frac{1}{81}\right)$	$e\left(\frac{119}{162}\right)$	$e\left(\frac{41}{81}\right)$

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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Modulus	$243$
Structure	\(C_{162}\)
Order	$162$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(11\)	\(13\)	\(14\)	\(16\)
\(\chi_{243}(1,\cdot)\)	243.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{243}(2,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{162}\right)\)	\(e\left(\frac{1}{81}\right)\)	\(e\left(\frac{23}{162}\right)\)	\(e\left(\frac{35}{81}\right)\)	\(e\left(\frac{1}{54}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{121}{162}\right)\)	\(e\left(\frac{4}{81}\right)\)	\(e\left(\frac{71}{162}\right)\)	\(e\left(\frac{2}{81}\right)\)
\(\chi_{243}(4,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{1}{81}\right)\)	\(e\left(\frac{2}{81}\right)\)	\(e\left(\frac{23}{81}\right)\)	\(e\left(\frac{70}{81}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{40}{81}\right)\)	\(e\left(\frac{8}{81}\right)\)	\(e\left(\frac{71}{81}\right)\)	\(e\left(\frac{4}{81}\right)\)
\(\chi_{243}(5,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{23}{162}\right)\)	\(e\left(\frac{23}{81}\right)\)	\(e\left(\frac{43}{162}\right)\)	\(e\left(\frac{76}{81}\right)\)	\(e\left(\frac{23}{54}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{29}{162}\right)\)	\(e\left(\frac{11}{81}\right)\)	\(e\left(\frac{13}{162}\right)\)	\(e\left(\frac{46}{81}\right)\)
\(\chi_{243}(7,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{35}{81}\right)\)	\(e\left(\frac{70}{81}\right)\)	\(e\left(\frac{76}{81}\right)\)	\(e\left(\frac{20}{81}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{23}{81}\right)\)	\(e\left(\frac{37}{81}\right)\)	\(e\left(\frac{55}{81}\right)\)	\(e\left(\frac{59}{81}\right)\)
\(\chi_{243}(8,\cdot)\)	243.h	54	no	\(-1\)	\(1\)	\(e\left(\frac{1}{54}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{23}{54}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{17}{54}\right)\)	\(e\left(\frac{2}{27}\right)\)
\(\chi_{243}(10,\cdot)\)	243.g	27	no	\(1\)	\(1\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)
\(\chi_{243}(11,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{121}{162}\right)\)	\(e\left(\frac{40}{81}\right)\)	\(e\left(\frac{29}{162}\right)\)	\(e\left(\frac{23}{81}\right)\)	\(e\left(\frac{13}{54}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{61}{162}\right)\)	\(e\left(\frac{79}{81}\right)\)	\(e\left(\frac{5}{162}\right)\)	\(e\left(\frac{80}{81}\right)\)
\(\chi_{243}(13,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{4}{81}\right)\)	\(e\left(\frac{8}{81}\right)\)	\(e\left(\frac{11}{81}\right)\)	\(e\left(\frac{37}{81}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{79}{81}\right)\)	\(e\left(\frac{32}{81}\right)\)	\(e\left(\frac{41}{81}\right)\)	\(e\left(\frac{16}{81}\right)\)
\(\chi_{243}(14,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{71}{162}\right)\)	\(e\left(\frac{71}{81}\right)\)	\(e\left(\frac{13}{162}\right)\)	\(e\left(\frac{55}{81}\right)\)	\(e\left(\frac{17}{54}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{5}{162}\right)\)	\(e\left(\frac{41}{81}\right)\)	\(e\left(\frac{19}{162}\right)\)	\(e\left(\frac{61}{81}\right)\)
\(\chi_{243}(16,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{2}{81}\right)\)	\(e\left(\frac{4}{81}\right)\)	\(e\left(\frac{46}{81}\right)\)	\(e\left(\frac{59}{81}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{80}{81}\right)\)	\(e\left(\frac{16}{81}\right)\)	\(e\left(\frac{61}{81}\right)\)	\(e\left(\frac{8}{81}\right)\)
\(\chi_{243}(17,\cdot)\)	243.h	54	no	\(-1\)	\(1\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{35}{54}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{22}{27}\right)\)
\(\chi_{243}(19,\cdot)\)	243.g	27	no	\(1\)	\(1\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{23}{27}\right)\)
\(\chi_{243}(20,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{25}{162}\right)\)	\(e\left(\frac{25}{81}\right)\)	\(e\left(\frac{89}{162}\right)\)	\(e\left(\frac{65}{81}\right)\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{109}{162}\right)\)	\(e\left(\frac{19}{81}\right)\)	\(e\left(\frac{155}{162}\right)\)	\(e\left(\frac{50}{81}\right)\)
\(\chi_{243}(22,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{61}{81}\right)\)	\(e\left(\frac{41}{81}\right)\)	\(e\left(\frac{26}{81}\right)\)	\(e\left(\frac{58}{81}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{10}{81}\right)\)	\(e\left(\frac{2}{81}\right)\)	\(e\left(\frac{38}{81}\right)\)	\(e\left(\frac{1}{81}\right)\)
\(\chi_{243}(23,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{65}{162}\right)\)	\(e\left(\frac{65}{81}\right)\)	\(e\left(\frac{37}{162}\right)\)	\(e\left(\frac{7}{81}\right)\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{89}{162}\right)\)	\(e\left(\frac{17}{81}\right)\)	\(e\left(\frac{79}{162}\right)\)	\(e\left(\frac{49}{81}\right)\)
\(\chi_{243}(25,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{23}{81}\right)\)	\(e\left(\frac{46}{81}\right)\)	\(e\left(\frac{43}{81}\right)\)	\(e\left(\frac{71}{81}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{29}{81}\right)\)	\(e\left(\frac{22}{81}\right)\)	\(e\left(\frac{13}{81}\right)\)	\(e\left(\frac{11}{81}\right)\)
\(\chi_{243}(26,\cdot)\)	243.f	18	no	\(-1\)	\(1\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{243}(28,\cdot)\)	243.e	9	no	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{243}(29,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{37}{162}\right)\)	\(e\left(\frac{37}{81}\right)\)	\(e\left(\frac{41}{162}\right)\)	\(e\left(\frac{80}{81}\right)\)	\(e\left(\frac{37}{54}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{103}{162}\right)\)	\(e\left(\frac{67}{81}\right)\)	\(e\left(\frac{35}{162}\right)\)	\(e\left(\frac{74}{81}\right)\)
\(\chi_{243}(31,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{10}{81}\right)\)	\(e\left(\frac{20}{81}\right)\)	\(e\left(\frac{68}{81}\right)\)	\(e\left(\frac{52}{81}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{26}{27}\right)\)	\(e\left(\frac{76}{81}\right)\)	\(e\left(\frac{80}{81}\right)\)	\(e\left(\frac{62}{81}\right)\)	\(e\left(\frac{40}{81}\right)\)
\(\chi_{243}(32,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{162}\right)\)	\(e\left(\frac{5}{81}\right)\)	\(e\left(\frac{115}{162}\right)\)	\(e\left(\frac{13}{81}\right)\)	\(e\left(\frac{5}{54}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{119}{162}\right)\)	\(e\left(\frac{20}{81}\right)\)	\(e\left(\frac{31}{162}\right)\)	\(e\left(\frac{10}{81}\right)\)
\(\chi_{243}(34,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{17}{81}\right)\)	\(e\left(\frac{34}{81}\right)\)	\(e\left(\frac{67}{81}\right)\)	\(e\left(\frac{56}{81}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{32}{81}\right)\)	\(e\left(\frac{55}{81}\right)\)	\(e\left(\frac{73}{81}\right)\)	\(e\left(\frac{68}{81}\right)\)
\(\chi_{243}(35,\cdot)\)	243.h	54	no	\(-1\)	\(1\)	\(e\left(\frac{31}{54}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{11}{54}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{8}{27}\right)\)
\(\chi_{243}(37,\cdot)\)	243.g	27	no	\(1\)	\(1\)	\(e\left(\frac{16}{27}\right)\)	\(e\left(\frac{5}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{19}{27}\right)\)	\(e\left(\frac{20}{27}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)
\(\chi_{243}(38,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{157}{162}\right)\)	\(e\left(\frac{76}{81}\right)\)	\(e\left(\frac{47}{162}\right)\)	\(e\left(\frac{68}{81}\right)\)	\(e\left(\frac{49}{54}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{43}{162}\right)\)	\(e\left(\frac{61}{81}\right)\)	\(e\left(\frac{131}{162}\right)\)	\(e\left(\frac{71}{81}\right)\)
\(\chi_{243}(40,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{13}{81}\right)\)	\(e\left(\frac{26}{81}\right)\)	\(e\left(\frac{56}{81}\right)\)	\(e\left(\frac{19}{81}\right)\)	\(e\left(\frac{13}{27}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{34}{81}\right)\)	\(e\left(\frac{23}{81}\right)\)	\(e\left(\frac{32}{81}\right)\)	\(e\left(\frac{52}{81}\right)\)
\(\chi_{243}(41,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{53}{162}\right)\)	\(e\left(\frac{53}{81}\right)\)	\(e\left(\frac{85}{162}\right)\)	\(e\left(\frac{73}{81}\right)\)	\(e\left(\frac{53}{54}\right)\)	\(e\left(\frac{23}{27}\right)\)	\(e\left(\frac{95}{162}\right)\)	\(e\left(\frac{50}{81}\right)\)	\(e\left(\frac{37}{162}\right)\)	\(e\left(\frac{25}{81}\right)\)
\(\chi_{243}(43,\cdot)\)	243.i	81	yes	\(1\)	\(1\)	\(e\left(\frac{65}{81}\right)\)	\(e\left(\frac{49}{81}\right)\)	\(e\left(\frac{37}{81}\right)\)	\(e\left(\frac{14}{81}\right)\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{8}{81}\right)\)	\(e\left(\frac{34}{81}\right)\)	\(e\left(\frac{79}{81}\right)\)	\(e\left(\frac{17}{81}\right)\)
\(\chi_{243}(44,\cdot)\)	243.h	54	no	\(-1\)	\(1\)	\(e\left(\frac{41}{54}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{25}{54}\right)\)	\(e\left(\frac{4}{27}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{47}{54}\right)\)	\(e\left(\frac{2}{27}\right)\)	\(e\left(\frac{49}{54}\right)\)	\(e\left(\frac{1}{27}\right)\)
\(\chi_{243}(46,\cdot)\)	243.g	27	no	\(1\)	\(1\)	\(e\left(\frac{11}{27}\right)\)	\(e\left(\frac{22}{27}\right)\)	\(e\left(\frac{10}{27}\right)\)	\(e\left(\frac{14}{27}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{8}{27}\right)\)	\(e\left(\frac{7}{27}\right)\)	\(e\left(\frac{25}{27}\right)\)	\(e\left(\frac{17}{27}\right)\)
\(\chi_{243}(47,\cdot)\)	243.j	162	yes	\(-1\)	\(1\)	\(e\left(\frac{61}{162}\right)\)	\(e\left(\frac{61}{81}\right)\)	\(e\left(\frac{107}{162}\right)\)	\(e\left(\frac{29}{81}\right)\)	\(e\left(\frac{7}{54}\right)\)	\(e\left(\frac{1}{27}\right)\)	\(e\left(\frac{91}{162}\right)\)	\(e\left(\frac{1}{81}\right)\)	\(e\left(\frac{119}{162}\right)\)	\(e\left(\frac{41}{81}\right)\)

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.