LMFDB - Group of Dirichlet characters of modulus 111

Show commands: Magma / Pari/GP / SageMath

comment:Define the Dirichlet group

sage:G = DirichletGroup(111)

gp:g = idealstar(,111,2)

magma:G = FullDirichletGroup(111);

Character group

Order	=	72	comment:Order sage:G.order() gp:g.no magma:Order(G);
Structure	=	$C_{2}\times C_{36}$	comment:Group structure sage:sorted(g.order() for g in G.gens()) gp:g.cyc magma:PrimaryInvariants(G);
Generators	=	$\chi_{111}(38,\cdot)$, $\chi_{111}(76,\cdot)$	comment:Generators sage:G.gens() gp:g.gen magma:Generators(G);

First 32 of 72 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	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$11$	$13$	$14$	$16$
$\chi_{111}(1,\cdot)$	111.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{111}(2,\cdot)$	111.q	36	yes	$1$	$1$	$e\left(\frac{19}{36}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{11}{36}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{111}(4,\cdot)$	111.o	18	no	$1$	$1$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{9}\right)$
$\chi_{111}(5,\cdot)$	111.q	36	yes	$1$	$1$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{111}(7,\cdot)$	111.k	9	no	$1$	$1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{111}(8,\cdot)$	111.m	12	yes	$1$	$1$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{3}\right)$	$-i$	$1$	$1$	$e\left(\frac{11}{12}\right)$	$i$	$e\left(\frac{1}{3}\right)$
$\chi_{111}(10,\cdot)$	111.e	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$
$\chi_{111}(11,\cdot)$	111.h	6	yes	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$-1$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$
$\chi_{111}(13,\cdot)$	111.r	36	no	$-1$	$1$	$e\left(\frac{11}{36}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{13}{36}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{2}{9}\right)$
$\chi_{111}(14,\cdot)$	111.m	12	yes	$1$	$1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{3}\right)$	$i$	$1$	$1$	$e\left(\frac{1}{12}\right)$	$-i$	$e\left(\frac{2}{3}\right)$
$\chi_{111}(16,\cdot)$	111.k	9	no	$1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{4}{9}\right)$
$\chi_{111}(17,\cdot)$	111.q	36	yes	$1$	$1$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{35}{36}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{111}(19,\cdot)$	111.r	36	no	$-1$	$1$	$e\left(\frac{35}{36}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{13}{36}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{111}(20,\cdot)$	111.q	36	yes	$1$	$1$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{111}(22,\cdot)$	111.r	36	no	$-1$	$1$	$e\left(\frac{31}{36}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{29}{36}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{4}{9}\right)$
$\chi_{111}(23,\cdot)$	111.m	12	yes	$1$	$1$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{3}\right)$	$-i$	$1$	$1$	$e\left(\frac{7}{12}\right)$	$i$	$e\left(\frac{2}{3}\right)$
$\chi_{111}(25,\cdot)$	111.o	18	no	$1$	$1$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{111}(26,\cdot)$	111.i	6	yes	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$
$\chi_{111}(28,\cdot)$	111.o	18	no	$1$	$1$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{111}(29,\cdot)$	111.m	12	yes	$1$	$1$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{2}{3}\right)$	$i$	$1$	$1$	$e\left(\frac{5}{12}\right)$	$-i$	$e\left(\frac{1}{3}\right)$
$\chi_{111}(31,\cdot)$	111.f	4	no	$-1$	$1$	$i$	$-1$	$-i$	$1$	$-i$	$1$	$-1$	$-i$	$i$	$1$
$\chi_{111}(32,\cdot)$	111.q	36	yes	$1$	$1$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{19}{36}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{111}(34,\cdot)$	111.k	9	no	$1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{111}(35,\cdot)$	111.q	36	yes	$1$	$1$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{29}{36}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{9}\right)$
$\chi_{111}(38,\cdot)$	111.b	2	no	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
$\chi_{111}(40,\cdot)$	111.o	18	no	$1$	$1$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{8}{9}\right)$
$\chi_{111}(41,\cdot)$	111.n	18	yes	$-1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{9}\right)$
$\chi_{111}(43,\cdot)$	111.f	4	no	$-1$	$1$	$-i$	$-1$	$i$	$1$	$i$	$1$	$-1$	$i$	$-i$	$1$
$\chi_{111}(44,\cdot)$	111.p	18	yes	$-1$	$1$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{9}\right)$
$\chi_{111}(46,\cdot)$	111.k	9	no	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{9}\right)$
$\chi_{111}(47,\cdot)$	111.i	6	yes	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$1$	$-1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{2}{3}\right)$
$\chi_{111}(49,\cdot)$	111.k	9	no	$1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{9}\right)$

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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	$111$
Structure	\(C_{2}\times C_{36}\)
Order	$72$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(11\)	\(13\)	\(14\)	\(16\)
\(\chi_{111}(1,\cdot)\)	111.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{111}(2,\cdot)\)	111.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{19}{36}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{36}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{111}(4,\cdot)\)	111.o	18	no	\(1\)	\(1\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{111}(5,\cdot)\)	111.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{111}(7,\cdot)\)	111.k	9	no	\(1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{111}(8,\cdot)\)	111.m	12	yes	\(1\)	\(1\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(1\)	\(1\)	\(e\left(\frac{11}{12}\right)\)	\(i\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{111}(10,\cdot)\)	111.e	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{111}(11,\cdot)\)	111.h	6	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{111}(13,\cdot)\)	111.r	36	no	\(-1\)	\(1\)	\(e\left(\frac{11}{36}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{13}{36}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{111}(14,\cdot)\)	111.m	12	yes	\(1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(1\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(-i\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{111}(16,\cdot)\)	111.k	9	no	\(1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{9}\right)\)
\(\chi_{111}(17,\cdot)\)	111.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{35}{36}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{111}(19,\cdot)\)	111.r	36	no	\(-1\)	\(1\)	\(e\left(\frac{35}{36}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{13}{36}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{111}(20,\cdot)\)	111.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{111}(22,\cdot)\)	111.r	36	no	\(-1\)	\(1\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{29}{36}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{4}{9}\right)\)
\(\chi_{111}(23,\cdot)\)	111.m	12	yes	\(1\)	\(1\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(1\)	\(1\)	\(e\left(\frac{7}{12}\right)\)	\(i\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{111}(25,\cdot)\)	111.o	18	no	\(1\)	\(1\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{111}(26,\cdot)\)	111.i	6	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{111}(28,\cdot)\)	111.o	18	no	\(1\)	\(1\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{111}(29,\cdot)\)	111.m	12	yes	\(1\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(i\)	\(1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(-i\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{111}(31,\cdot)\)	111.f	4	no	\(-1\)	\(1\)	\(i\)	\(-1\)	\(-i\)	\(1\)	\(-i\)	\(1\)	\(-1\)	\(-i\)	\(i\)	\(1\)
\(\chi_{111}(32,\cdot)\)	111.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{19}{36}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{111}(34,\cdot)\)	111.k	9	no	\(1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{111}(35,\cdot)\)	111.q	36	yes	\(1\)	\(1\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{29}{36}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{111}(38,\cdot)\)	111.b	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)
\(\chi_{111}(40,\cdot)\)	111.o	18	no	\(1\)	\(1\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{8}{9}\right)\)
\(\chi_{111}(41,\cdot)\)	111.n	18	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{111}(43,\cdot)\)	111.f	4	no	\(-1\)	\(1\)	\(-i\)	\(-1\)	\(i\)	\(1\)	\(i\)	\(1\)	\(-1\)	\(i\)	\(-i\)	\(1\)
\(\chi_{111}(44,\cdot)\)	111.p	18	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{9}\right)\)
\(\chi_{111}(46,\cdot)\)	111.k	9	no	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{9}\right)\)
\(\chi_{111}(47,\cdot)\)	111.i	6	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(1\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{111}(49,\cdot)\)	111.k	9	no	\(1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{9}\right)\)

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First 32 of 72 characters

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