LMFDB - Group of Dirichlet characters of modulus 269

Show commands: PariGP / SageMath

sage: H = DirichletGroup(269)

pari: g = idealstar(,269,2)

Character group

sage: G.order() pari: g.no
Order	=	268
sage: H.invariants() pari: g.cyc
Structure	=	$C_{268}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{269}(2,\cdot)$

First 32 of 268 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{269}(1,\cdot)$	269.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{269}(2,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{1}{268}\right)$	$e\left(\frac{109}{268}\right)$	$e\left(\frac{1}{134}\right)$	$e\left(\frac{52}{67}\right)$	$e\left(\frac{55}{134}\right)$	$e\left(\frac{19}{268}\right)$	$e\left(\frac{3}{268}\right)$	$e\left(\frac{109}{134}\right)$	$e\left(\frac{209}{268}\right)$	$e\left(\frac{115}{134}\right)$
$\chi_{269}(3,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{109}{268}\right)$	$e\left(\frac{89}{268}\right)$	$e\left(\frac{109}{134}\right)$	$e\left(\frac{40}{67}\right)$	$e\left(\frac{99}{134}\right)$	$e\left(\frac{195}{268}\right)$	$e\left(\frac{59}{268}\right)$	$e\left(\frac{89}{134}\right)$	$e\left(\frac{1}{268}\right)$	$e\left(\frac{73}{134}\right)$
$\chi_{269}(4,\cdot)$	269.e	134	yes	$1$	$1$	$e\left(\frac{1}{134}\right)$	$e\left(\frac{109}{134}\right)$	$e\left(\frac{1}{67}\right)$	$e\left(\frac{37}{67}\right)$	$e\left(\frac{55}{67}\right)$	$e\left(\frac{19}{134}\right)$	$e\left(\frac{3}{134}\right)$	$e\left(\frac{42}{67}\right)$	$e\left(\frac{75}{134}\right)$	$e\left(\frac{48}{67}\right)$
$\chi_{269}(5,\cdot)$	269.d	67	yes	$1$	$1$	$e\left(\frac{52}{67}\right)$	$e\left(\frac{40}{67}\right)$	$e\left(\frac{37}{67}\right)$	$e\left(\frac{29}{67}\right)$	$e\left(\frac{25}{67}\right)$	$e\left(\frac{50}{67}\right)$	$e\left(\frac{22}{67}\right)$	$e\left(\frac{13}{67}\right)$	$e\left(\frac{14}{67}\right)$	$e\left(\frac{34}{67}\right)$
$\chi_{269}(6,\cdot)$	269.e	134	yes	$1$	$1$	$e\left(\frac{55}{134}\right)$	$e\left(\frac{99}{134}\right)$	$e\left(\frac{55}{67}\right)$	$e\left(\frac{25}{67}\right)$	$e\left(\frac{10}{67}\right)$	$e\left(\frac{107}{134}\right)$	$e\left(\frac{31}{134}\right)$	$e\left(\frac{32}{67}\right)$	$e\left(\frac{105}{134}\right)$	$e\left(\frac{27}{67}\right)$
$\chi_{269}(7,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{19}{268}\right)$	$e\left(\frac{195}{268}\right)$	$e\left(\frac{19}{134}\right)$	$e\left(\frac{50}{67}\right)$	$e\left(\frac{107}{134}\right)$	$e\left(\frac{93}{268}\right)$	$e\left(\frac{57}{268}\right)$	$e\left(\frac{61}{134}\right)$	$e\left(\frac{219}{268}\right)$	$e\left(\frac{41}{134}\right)$
$\chi_{269}(8,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{3}{268}\right)$	$e\left(\frac{59}{268}\right)$	$e\left(\frac{3}{134}\right)$	$e\left(\frac{22}{67}\right)$	$e\left(\frac{31}{134}\right)$	$e\left(\frac{57}{268}\right)$	$e\left(\frac{9}{268}\right)$	$e\left(\frac{59}{134}\right)$	$e\left(\frac{91}{268}\right)$	$e\left(\frac{77}{134}\right)$
$\chi_{269}(9,\cdot)$	269.e	134	yes	$1$	$1$	$e\left(\frac{109}{134}\right)$	$e\left(\frac{89}{134}\right)$	$e\left(\frac{42}{67}\right)$	$e\left(\frac{13}{67}\right)$	$e\left(\frac{32}{67}\right)$	$e\left(\frac{61}{134}\right)$	$e\left(\frac{59}{134}\right)$	$e\left(\frac{22}{67}\right)$	$e\left(\frac{1}{134}\right)$	$e\left(\frac{6}{67}\right)$
$\chi_{269}(10,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{209}{268}\right)$	$e\left(\frac{1}{268}\right)$	$e\left(\frac{75}{134}\right)$	$e\left(\frac{14}{67}\right)$	$e\left(\frac{105}{134}\right)$	$e\left(\frac{219}{268}\right)$	$e\left(\frac{91}{268}\right)$	$e\left(\frac{1}{134}\right)$	$e\left(\frac{265}{268}\right)$	$e\left(\frac{49}{134}\right)$
$\chi_{269}(11,\cdot)$	269.e	134	yes	$1$	$1$	$e\left(\frac{115}{134}\right)$	$e\left(\frac{73}{134}\right)$	$e\left(\frac{48}{67}\right)$	$e\left(\frac{34}{67}\right)$	$e\left(\frac{27}{67}\right)$	$e\left(\frac{41}{134}\right)$	$e\left(\frac{77}{134}\right)$	$e\left(\frac{6}{67}\right)$	$e\left(\frac{49}{134}\right)$	$e\left(\frac{26}{67}\right)$
$\chi_{269}(12,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{111}{268}\right)$	$e\left(\frac{39}{268}\right)$	$e\left(\frac{111}{134}\right)$	$e\left(\frac{10}{67}\right)$	$e\left(\frac{75}{134}\right)$	$e\left(\frac{233}{268}\right)$	$e\left(\frac{65}{268}\right)$	$e\left(\frac{39}{134}\right)$	$e\left(\frac{151}{268}\right)$	$e\left(\frac{35}{134}\right)$
$\chi_{269}(13,\cdot)$	269.e	134	yes	$1$	$1$	$e\left(\frac{71}{134}\right)$	$e\left(\frac{101}{134}\right)$	$e\left(\frac{4}{67}\right)$	$e\left(\frac{14}{67}\right)$	$e\left(\frac{19}{67}\right)$	$e\left(\frac{9}{134}\right)$	$e\left(\frac{79}{134}\right)$	$e\left(\frac{34}{67}\right)$	$e\left(\frac{99}{134}\right)$	$e\left(\frac{58}{67}\right)$
$\chi_{269}(14,\cdot)$	269.d	67	yes	$1$	$1$	$e\left(\frac{5}{67}\right)$	$e\left(\frac{9}{67}\right)$	$e\left(\frac{10}{67}\right)$	$e\left(\frac{35}{67}\right)$	$e\left(\frac{14}{67}\right)$	$e\left(\frac{28}{67}\right)$	$e\left(\frac{15}{67}\right)$	$e\left(\frac{18}{67}\right)$	$e\left(\frac{40}{67}\right)$	$e\left(\frac{11}{67}\right)$
$\chi_{269}(15,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{49}{268}\right)$	$e\left(\frac{249}{268}\right)$	$e\left(\frac{49}{134}\right)$	$e\left(\frac{2}{67}\right)$	$e\left(\frac{15}{134}\right)$	$e\left(\frac{127}{268}\right)$	$e\left(\frac{147}{268}\right)$	$e\left(\frac{115}{134}\right)$	$e\left(\frac{57}{268}\right)$	$e\left(\frac{7}{134}\right)$
$\chi_{269}(16,\cdot)$	269.d	67	yes	$1$	$1$	$e\left(\frac{1}{67}\right)$	$e\left(\frac{42}{67}\right)$	$e\left(\frac{2}{67}\right)$	$e\left(\frac{7}{67}\right)$	$e\left(\frac{43}{67}\right)$	$e\left(\frac{19}{67}\right)$	$e\left(\frac{3}{67}\right)$	$e\left(\frac{17}{67}\right)$	$e\left(\frac{8}{67}\right)$	$e\left(\frac{29}{67}\right)$
$\chi_{269}(17,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{105}{268}\right)$	$e\left(\frac{189}{268}\right)$	$e\left(\frac{105}{134}\right)$	$e\left(\frac{33}{67}\right)$	$e\left(\frac{13}{134}\right)$	$e\left(\frac{119}{268}\right)$	$e\left(\frac{47}{268}\right)$	$e\left(\frac{55}{134}\right)$	$e\left(\frac{237}{268}\right)$	$e\left(\frac{15}{134}\right)$
$\chi_{269}(18,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{219}{268}\right)$	$e\left(\frac{19}{268}\right)$	$e\left(\frac{85}{134}\right)$	$e\left(\frac{65}{67}\right)$	$e\left(\frac{119}{134}\right)$	$e\left(\frac{141}{268}\right)$	$e\left(\frac{121}{268}\right)$	$e\left(\frac{19}{134}\right)$	$e\left(\frac{211}{268}\right)$	$e\left(\frac{127}{134}\right)$
$\chi_{269}(19,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{223}{268}\right)$	$e\left(\frac{187}{268}\right)$	$e\left(\frac{89}{134}\right)$	$e\left(\frac{5}{67}\right)$	$e\left(\frac{71}{134}\right)$	$e\left(\frac{217}{268}\right)$	$e\left(\frac{133}{268}\right)$	$e\left(\frac{53}{134}\right)$	$e\left(\frac{243}{268}\right)$	$e\left(\frac{51}{134}\right)$
$\chi_{269}(20,\cdot)$	269.e	134	yes	$1$	$1$	$e\left(\frac{105}{134}\right)$	$e\left(\frac{55}{134}\right)$	$e\left(\frac{38}{67}\right)$	$e\left(\frac{66}{67}\right)$	$e\left(\frac{13}{67}\right)$	$e\left(\frac{119}{134}\right)$	$e\left(\frac{47}{134}\right)$	$e\left(\frac{55}{67}\right)$	$e\left(\frac{103}{134}\right)$	$e\left(\frac{15}{67}\right)$
$\chi_{269}(21,\cdot)$	269.d	67	yes	$1$	$1$	$e\left(\frac{32}{67}\right)$	$e\left(\frac{4}{67}\right)$	$e\left(\frac{64}{67}\right)$	$e\left(\frac{23}{67}\right)$	$e\left(\frac{36}{67}\right)$	$e\left(\frac{5}{67}\right)$	$e\left(\frac{29}{67}\right)$	$e\left(\frac{8}{67}\right)$	$e\left(\frac{55}{67}\right)$	$e\left(\frac{57}{67}\right)$
$\chi_{269}(22,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{231}{268}\right)$	$e\left(\frac{255}{268}\right)$	$e\left(\frac{97}{134}\right)$	$e\left(\frac{19}{67}\right)$	$e\left(\frac{109}{134}\right)$	$e\left(\frac{101}{268}\right)$	$e\left(\frac{157}{268}\right)$	$e\left(\frac{121}{134}\right)$	$e\left(\frac{39}{268}\right)$	$e\left(\frac{33}{134}\right)$
$\chi_{269}(23,\cdot)$	269.d	67	yes	$1$	$1$	$e\left(\frac{44}{67}\right)$	$e\left(\frac{39}{67}\right)$	$e\left(\frac{21}{67}\right)$	$e\left(\frac{40}{67}\right)$	$e\left(\frac{16}{67}\right)$	$e\left(\frac{32}{67}\right)$	$e\left(\frac{65}{67}\right)$	$e\left(\frac{11}{67}\right)$	$e\left(\frac{17}{67}\right)$	$e\left(\frac{3}{67}\right)$
$\chi_{269}(24,\cdot)$	269.d	67	yes	$1$	$1$	$e\left(\frac{28}{67}\right)$	$e\left(\frac{37}{67}\right)$	$e\left(\frac{56}{67}\right)$	$e\left(\frac{62}{67}\right)$	$e\left(\frac{65}{67}\right)$	$e\left(\frac{63}{67}\right)$	$e\left(\frac{17}{67}\right)$	$e\left(\frac{7}{67}\right)$	$e\left(\frac{23}{67}\right)$	$e\left(\frac{8}{67}\right)$
$\chi_{269}(25,\cdot)$	269.d	67	yes	$1$	$1$	$e\left(\frac{37}{67}\right)$	$e\left(\frac{13}{67}\right)$	$e\left(\frac{7}{67}\right)$	$e\left(\frac{58}{67}\right)$	$e\left(\frac{50}{67}\right)$	$e\left(\frac{33}{67}\right)$	$e\left(\frac{44}{67}\right)$	$e\left(\frac{26}{67}\right)$	$e\left(\frac{28}{67}\right)$	$e\left(\frac{1}{67}\right)$
$\chi_{269}(26,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{143}{268}\right)$	$e\left(\frac{43}{268}\right)$	$e\left(\frac{9}{134}\right)$	$e\left(\frac{66}{67}\right)$	$e\left(\frac{93}{134}\right)$	$e\left(\frac{37}{268}\right)$	$e\left(\frac{161}{268}\right)$	$e\left(\frac{43}{134}\right)$	$e\left(\frac{139}{268}\right)$	$e\left(\frac{97}{134}\right)$
$\chi_{269}(27,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{59}{268}\right)$	$e\left(\frac{267}{268}\right)$	$e\left(\frac{59}{134}\right)$	$e\left(\frac{53}{67}\right)$	$e\left(\frac{29}{134}\right)$	$e\left(\frac{49}{268}\right)$	$e\left(\frac{177}{268}\right)$	$e\left(\frac{133}{134}\right)$	$e\left(\frac{3}{268}\right)$	$e\left(\frac{85}{134}\right)$
$\chi_{269}(28,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{21}{268}\right)$	$e\left(\frac{145}{268}\right)$	$e\left(\frac{21}{134}\right)$	$e\left(\frac{20}{67}\right)$	$e\left(\frac{83}{134}\right)$	$e\left(\frac{131}{268}\right)$	$e\left(\frac{63}{268}\right)$	$e\left(\frac{11}{134}\right)$	$e\left(\frac{101}{268}\right)$	$e\left(\frac{3}{134}\right)$
$\chi_{269}(29,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{187}{268}\right)$	$e\left(\frac{15}{268}\right)$	$e\left(\frac{53}{134}\right)$	$e\left(\frac{9}{67}\right)$	$e\left(\frac{101}{134}\right)$	$e\left(\frac{69}{268}\right)$	$e\left(\frac{25}{268}\right)$	$e\left(\frac{15}{134}\right)$	$e\left(\frac{223}{268}\right)$	$e\left(\frac{65}{134}\right)$
$\chi_{269}(30,\cdot)$	269.e	134	yes	$1$	$1$	$e\left(\frac{25}{134}\right)$	$e\left(\frac{45}{134}\right)$	$e\left(\frac{25}{67}\right)$	$e\left(\frac{54}{67}\right)$	$e\left(\frac{35}{67}\right)$	$e\left(\frac{73}{134}\right)$	$e\left(\frac{75}{134}\right)$	$e\left(\frac{45}{67}\right)$	$e\left(\frac{133}{134}\right)$	$e\left(\frac{61}{67}\right)$
$\chi_{269}(31,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{259}{268}\right)$	$e\left(\frac{91}{268}\right)$	$e\left(\frac{125}{134}\right)$	$e\left(\frac{1}{67}\right)$	$e\left(\frac{41}{134}\right)$	$e\left(\frac{97}{268}\right)$	$e\left(\frac{241}{268}\right)$	$e\left(\frac{91}{134}\right)$	$e\left(\frac{263}{268}\right)$	$e\left(\frac{37}{134}\right)$
$\chi_{269}(32,\cdot)$	269.f	268	yes	$-1$	$1$	$e\left(\frac{5}{268}\right)$	$e\left(\frac{9}{268}\right)$	$e\left(\frac{5}{134}\right)$	$e\left(\frac{59}{67}\right)$	$e\left(\frac{7}{134}\right)$	$e\left(\frac{95}{268}\right)$	$e\left(\frac{15}{268}\right)$	$e\left(\frac{9}{134}\right)$	$e\left(\frac{241}{268}\right)$	$e\left(\frac{39}{134}\right)$

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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 $\F_{q}$

Modulus	$269$
Structure	\(C_{268}\)
Order	$268$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{269}(1,\cdot)\)	269.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{269}(2,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{268}\right)\)	\(e\left(\frac{109}{268}\right)\)	\(e\left(\frac{1}{134}\right)\)	\(e\left(\frac{52}{67}\right)\)	\(e\left(\frac{55}{134}\right)\)	\(e\left(\frac{19}{268}\right)\)	\(e\left(\frac{3}{268}\right)\)	\(e\left(\frac{109}{134}\right)\)	\(e\left(\frac{209}{268}\right)\)	\(e\left(\frac{115}{134}\right)\)
\(\chi_{269}(3,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{109}{268}\right)\)	\(e\left(\frac{89}{268}\right)\)	\(e\left(\frac{109}{134}\right)\)	\(e\left(\frac{40}{67}\right)\)	\(e\left(\frac{99}{134}\right)\)	\(e\left(\frac{195}{268}\right)\)	\(e\left(\frac{59}{268}\right)\)	\(e\left(\frac{89}{134}\right)\)	\(e\left(\frac{1}{268}\right)\)	\(e\left(\frac{73}{134}\right)\)
\(\chi_{269}(4,\cdot)\)	269.e	134	yes	\(1\)	\(1\)	\(e\left(\frac{1}{134}\right)\)	\(e\left(\frac{109}{134}\right)\)	\(e\left(\frac{1}{67}\right)\)	\(e\left(\frac{37}{67}\right)\)	\(e\left(\frac{55}{67}\right)\)	\(e\left(\frac{19}{134}\right)\)	\(e\left(\frac{3}{134}\right)\)	\(e\left(\frac{42}{67}\right)\)	\(e\left(\frac{75}{134}\right)\)	\(e\left(\frac{48}{67}\right)\)
\(\chi_{269}(5,\cdot)\)	269.d	67	yes	\(1\)	\(1\)	\(e\left(\frac{52}{67}\right)\)	\(e\left(\frac{40}{67}\right)\)	\(e\left(\frac{37}{67}\right)\)	\(e\left(\frac{29}{67}\right)\)	\(e\left(\frac{25}{67}\right)\)	\(e\left(\frac{50}{67}\right)\)	\(e\left(\frac{22}{67}\right)\)	\(e\left(\frac{13}{67}\right)\)	\(e\left(\frac{14}{67}\right)\)	\(e\left(\frac{34}{67}\right)\)
\(\chi_{269}(6,\cdot)\)	269.e	134	yes	\(1\)	\(1\)	\(e\left(\frac{55}{134}\right)\)	\(e\left(\frac{99}{134}\right)\)	\(e\left(\frac{55}{67}\right)\)	\(e\left(\frac{25}{67}\right)\)	\(e\left(\frac{10}{67}\right)\)	\(e\left(\frac{107}{134}\right)\)	\(e\left(\frac{31}{134}\right)\)	\(e\left(\frac{32}{67}\right)\)	\(e\left(\frac{105}{134}\right)\)	\(e\left(\frac{27}{67}\right)\)
\(\chi_{269}(7,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{268}\right)\)	\(e\left(\frac{195}{268}\right)\)	\(e\left(\frac{19}{134}\right)\)	\(e\left(\frac{50}{67}\right)\)	\(e\left(\frac{107}{134}\right)\)	\(e\left(\frac{93}{268}\right)\)	\(e\left(\frac{57}{268}\right)\)	\(e\left(\frac{61}{134}\right)\)	\(e\left(\frac{219}{268}\right)\)	\(e\left(\frac{41}{134}\right)\)
\(\chi_{269}(8,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{268}\right)\)	\(e\left(\frac{59}{268}\right)\)	\(e\left(\frac{3}{134}\right)\)	\(e\left(\frac{22}{67}\right)\)	\(e\left(\frac{31}{134}\right)\)	\(e\left(\frac{57}{268}\right)\)	\(e\left(\frac{9}{268}\right)\)	\(e\left(\frac{59}{134}\right)\)	\(e\left(\frac{91}{268}\right)\)	\(e\left(\frac{77}{134}\right)\)
\(\chi_{269}(9,\cdot)\)	269.e	134	yes	\(1\)	\(1\)	\(e\left(\frac{109}{134}\right)\)	\(e\left(\frac{89}{134}\right)\)	\(e\left(\frac{42}{67}\right)\)	\(e\left(\frac{13}{67}\right)\)	\(e\left(\frac{32}{67}\right)\)	\(e\left(\frac{61}{134}\right)\)	\(e\left(\frac{59}{134}\right)\)	\(e\left(\frac{22}{67}\right)\)	\(e\left(\frac{1}{134}\right)\)	\(e\left(\frac{6}{67}\right)\)
\(\chi_{269}(10,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{209}{268}\right)\)	\(e\left(\frac{1}{268}\right)\)	\(e\left(\frac{75}{134}\right)\)	\(e\left(\frac{14}{67}\right)\)	\(e\left(\frac{105}{134}\right)\)	\(e\left(\frac{219}{268}\right)\)	\(e\left(\frac{91}{268}\right)\)	\(e\left(\frac{1}{134}\right)\)	\(e\left(\frac{265}{268}\right)\)	\(e\left(\frac{49}{134}\right)\)
\(\chi_{269}(11,\cdot)\)	269.e	134	yes	\(1\)	\(1\)	\(e\left(\frac{115}{134}\right)\)	\(e\left(\frac{73}{134}\right)\)	\(e\left(\frac{48}{67}\right)\)	\(e\left(\frac{34}{67}\right)\)	\(e\left(\frac{27}{67}\right)\)	\(e\left(\frac{41}{134}\right)\)	\(e\left(\frac{77}{134}\right)\)	\(e\left(\frac{6}{67}\right)\)	\(e\left(\frac{49}{134}\right)\)	\(e\left(\frac{26}{67}\right)\)
\(\chi_{269}(12,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{111}{268}\right)\)	\(e\left(\frac{39}{268}\right)\)	\(e\left(\frac{111}{134}\right)\)	\(e\left(\frac{10}{67}\right)\)	\(e\left(\frac{75}{134}\right)\)	\(e\left(\frac{233}{268}\right)\)	\(e\left(\frac{65}{268}\right)\)	\(e\left(\frac{39}{134}\right)\)	\(e\left(\frac{151}{268}\right)\)	\(e\left(\frac{35}{134}\right)\)
\(\chi_{269}(13,\cdot)\)	269.e	134	yes	\(1\)	\(1\)	\(e\left(\frac{71}{134}\right)\)	\(e\left(\frac{101}{134}\right)\)	\(e\left(\frac{4}{67}\right)\)	\(e\left(\frac{14}{67}\right)\)	\(e\left(\frac{19}{67}\right)\)	\(e\left(\frac{9}{134}\right)\)	\(e\left(\frac{79}{134}\right)\)	\(e\left(\frac{34}{67}\right)\)	\(e\left(\frac{99}{134}\right)\)	\(e\left(\frac{58}{67}\right)\)
\(\chi_{269}(14,\cdot)\)	269.d	67	yes	\(1\)	\(1\)	\(e\left(\frac{5}{67}\right)\)	\(e\left(\frac{9}{67}\right)\)	\(e\left(\frac{10}{67}\right)\)	\(e\left(\frac{35}{67}\right)\)	\(e\left(\frac{14}{67}\right)\)	\(e\left(\frac{28}{67}\right)\)	\(e\left(\frac{15}{67}\right)\)	\(e\left(\frac{18}{67}\right)\)	\(e\left(\frac{40}{67}\right)\)	\(e\left(\frac{11}{67}\right)\)
\(\chi_{269}(15,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{49}{268}\right)\)	\(e\left(\frac{249}{268}\right)\)	\(e\left(\frac{49}{134}\right)\)	\(e\left(\frac{2}{67}\right)\)	\(e\left(\frac{15}{134}\right)\)	\(e\left(\frac{127}{268}\right)\)	\(e\left(\frac{147}{268}\right)\)	\(e\left(\frac{115}{134}\right)\)	\(e\left(\frac{57}{268}\right)\)	\(e\left(\frac{7}{134}\right)\)
\(\chi_{269}(16,\cdot)\)	269.d	67	yes	\(1\)	\(1\)	\(e\left(\frac{1}{67}\right)\)	\(e\left(\frac{42}{67}\right)\)	\(e\left(\frac{2}{67}\right)\)	\(e\left(\frac{7}{67}\right)\)	\(e\left(\frac{43}{67}\right)\)	\(e\left(\frac{19}{67}\right)\)	\(e\left(\frac{3}{67}\right)\)	\(e\left(\frac{17}{67}\right)\)	\(e\left(\frac{8}{67}\right)\)	\(e\left(\frac{29}{67}\right)\)
\(\chi_{269}(17,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{105}{268}\right)\)	\(e\left(\frac{189}{268}\right)\)	\(e\left(\frac{105}{134}\right)\)	\(e\left(\frac{33}{67}\right)\)	\(e\left(\frac{13}{134}\right)\)	\(e\left(\frac{119}{268}\right)\)	\(e\left(\frac{47}{268}\right)\)	\(e\left(\frac{55}{134}\right)\)	\(e\left(\frac{237}{268}\right)\)	\(e\left(\frac{15}{134}\right)\)
\(\chi_{269}(18,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{219}{268}\right)\)	\(e\left(\frac{19}{268}\right)\)	\(e\left(\frac{85}{134}\right)\)	\(e\left(\frac{65}{67}\right)\)	\(e\left(\frac{119}{134}\right)\)	\(e\left(\frac{141}{268}\right)\)	\(e\left(\frac{121}{268}\right)\)	\(e\left(\frac{19}{134}\right)\)	\(e\left(\frac{211}{268}\right)\)	\(e\left(\frac{127}{134}\right)\)
\(\chi_{269}(19,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{223}{268}\right)\)	\(e\left(\frac{187}{268}\right)\)	\(e\left(\frac{89}{134}\right)\)	\(e\left(\frac{5}{67}\right)\)	\(e\left(\frac{71}{134}\right)\)	\(e\left(\frac{217}{268}\right)\)	\(e\left(\frac{133}{268}\right)\)	\(e\left(\frac{53}{134}\right)\)	\(e\left(\frac{243}{268}\right)\)	\(e\left(\frac{51}{134}\right)\)
\(\chi_{269}(20,\cdot)\)	269.e	134	yes	\(1\)	\(1\)	\(e\left(\frac{105}{134}\right)\)	\(e\left(\frac{55}{134}\right)\)	\(e\left(\frac{38}{67}\right)\)	\(e\left(\frac{66}{67}\right)\)	\(e\left(\frac{13}{67}\right)\)	\(e\left(\frac{119}{134}\right)\)	\(e\left(\frac{47}{134}\right)\)	\(e\left(\frac{55}{67}\right)\)	\(e\left(\frac{103}{134}\right)\)	\(e\left(\frac{15}{67}\right)\)
\(\chi_{269}(21,\cdot)\)	269.d	67	yes	\(1\)	\(1\)	\(e\left(\frac{32}{67}\right)\)	\(e\left(\frac{4}{67}\right)\)	\(e\left(\frac{64}{67}\right)\)	\(e\left(\frac{23}{67}\right)\)	\(e\left(\frac{36}{67}\right)\)	\(e\left(\frac{5}{67}\right)\)	\(e\left(\frac{29}{67}\right)\)	\(e\left(\frac{8}{67}\right)\)	\(e\left(\frac{55}{67}\right)\)	\(e\left(\frac{57}{67}\right)\)
\(\chi_{269}(22,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{231}{268}\right)\)	\(e\left(\frac{255}{268}\right)\)	\(e\left(\frac{97}{134}\right)\)	\(e\left(\frac{19}{67}\right)\)	\(e\left(\frac{109}{134}\right)\)	\(e\left(\frac{101}{268}\right)\)	\(e\left(\frac{157}{268}\right)\)	\(e\left(\frac{121}{134}\right)\)	\(e\left(\frac{39}{268}\right)\)	\(e\left(\frac{33}{134}\right)\)
\(\chi_{269}(23,\cdot)\)	269.d	67	yes	\(1\)	\(1\)	\(e\left(\frac{44}{67}\right)\)	\(e\left(\frac{39}{67}\right)\)	\(e\left(\frac{21}{67}\right)\)	\(e\left(\frac{40}{67}\right)\)	\(e\left(\frac{16}{67}\right)\)	\(e\left(\frac{32}{67}\right)\)	\(e\left(\frac{65}{67}\right)\)	\(e\left(\frac{11}{67}\right)\)	\(e\left(\frac{17}{67}\right)\)	\(e\left(\frac{3}{67}\right)\)
\(\chi_{269}(24,\cdot)\)	269.d	67	yes	\(1\)	\(1\)	\(e\left(\frac{28}{67}\right)\)	\(e\left(\frac{37}{67}\right)\)	\(e\left(\frac{56}{67}\right)\)	\(e\left(\frac{62}{67}\right)\)	\(e\left(\frac{65}{67}\right)\)	\(e\left(\frac{63}{67}\right)\)	\(e\left(\frac{17}{67}\right)\)	\(e\left(\frac{7}{67}\right)\)	\(e\left(\frac{23}{67}\right)\)	\(e\left(\frac{8}{67}\right)\)
\(\chi_{269}(25,\cdot)\)	269.d	67	yes	\(1\)	\(1\)	\(e\left(\frac{37}{67}\right)\)	\(e\left(\frac{13}{67}\right)\)	\(e\left(\frac{7}{67}\right)\)	\(e\left(\frac{58}{67}\right)\)	\(e\left(\frac{50}{67}\right)\)	\(e\left(\frac{33}{67}\right)\)	\(e\left(\frac{44}{67}\right)\)	\(e\left(\frac{26}{67}\right)\)	\(e\left(\frac{28}{67}\right)\)	\(e\left(\frac{1}{67}\right)\)
\(\chi_{269}(26,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{143}{268}\right)\)	\(e\left(\frac{43}{268}\right)\)	\(e\left(\frac{9}{134}\right)\)	\(e\left(\frac{66}{67}\right)\)	\(e\left(\frac{93}{134}\right)\)	\(e\left(\frac{37}{268}\right)\)	\(e\left(\frac{161}{268}\right)\)	\(e\left(\frac{43}{134}\right)\)	\(e\left(\frac{139}{268}\right)\)	\(e\left(\frac{97}{134}\right)\)
\(\chi_{269}(27,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{59}{268}\right)\)	\(e\left(\frac{267}{268}\right)\)	\(e\left(\frac{59}{134}\right)\)	\(e\left(\frac{53}{67}\right)\)	\(e\left(\frac{29}{134}\right)\)	\(e\left(\frac{49}{268}\right)\)	\(e\left(\frac{177}{268}\right)\)	\(e\left(\frac{133}{134}\right)\)	\(e\left(\frac{3}{268}\right)\)	\(e\left(\frac{85}{134}\right)\)
\(\chi_{269}(28,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{21}{268}\right)\)	\(e\left(\frac{145}{268}\right)\)	\(e\left(\frac{21}{134}\right)\)	\(e\left(\frac{20}{67}\right)\)	\(e\left(\frac{83}{134}\right)\)	\(e\left(\frac{131}{268}\right)\)	\(e\left(\frac{63}{268}\right)\)	\(e\left(\frac{11}{134}\right)\)	\(e\left(\frac{101}{268}\right)\)	\(e\left(\frac{3}{134}\right)\)
\(\chi_{269}(29,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{187}{268}\right)\)	\(e\left(\frac{15}{268}\right)\)	\(e\left(\frac{53}{134}\right)\)	\(e\left(\frac{9}{67}\right)\)	\(e\left(\frac{101}{134}\right)\)	\(e\left(\frac{69}{268}\right)\)	\(e\left(\frac{25}{268}\right)\)	\(e\left(\frac{15}{134}\right)\)	\(e\left(\frac{223}{268}\right)\)	\(e\left(\frac{65}{134}\right)\)
\(\chi_{269}(30,\cdot)\)	269.e	134	yes	\(1\)	\(1\)	\(e\left(\frac{25}{134}\right)\)	\(e\left(\frac{45}{134}\right)\)	\(e\left(\frac{25}{67}\right)\)	\(e\left(\frac{54}{67}\right)\)	\(e\left(\frac{35}{67}\right)\)	\(e\left(\frac{73}{134}\right)\)	\(e\left(\frac{75}{134}\right)\)	\(e\left(\frac{45}{67}\right)\)	\(e\left(\frac{133}{134}\right)\)	\(e\left(\frac{61}{67}\right)\)
\(\chi_{269}(31,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{259}{268}\right)\)	\(e\left(\frac{91}{268}\right)\)	\(e\left(\frac{125}{134}\right)\)	\(e\left(\frac{1}{67}\right)\)	\(e\left(\frac{41}{134}\right)\)	\(e\left(\frac{97}{268}\right)\)	\(e\left(\frac{241}{268}\right)\)	\(e\left(\frac{91}{134}\right)\)	\(e\left(\frac{263}{268}\right)\)	\(e\left(\frac{37}{134}\right)\)
\(\chi_{269}(32,\cdot)\)	269.f	268	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{268}\right)\)	\(e\left(\frac{9}{268}\right)\)	\(e\left(\frac{5}{134}\right)\)	\(e\left(\frac{59}{67}\right)\)	\(e\left(\frac{7}{134}\right)\)	\(e\left(\frac{95}{268}\right)\)	\(e\left(\frac{15}{268}\right)\)	\(e\left(\frac{9}{134}\right)\)	\(e\left(\frac{241}{268}\right)\)	\(e\left(\frac{39}{134}\right)\)

Introduction

L-functions

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Character group

First 32 of 268 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.