# Properties

 Modulus $571$ Structure $$C_{570}$$ Order $570$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(571)

pari: g = idealstar(,571,2)

## Character group

 sage: G.order()  pari: g.no Order = 570 sage: H.invariants()  pari: g.cyc Structure = $$C_{570}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{571}(3,\cdot)$

## First 32 of 570 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{571}(1,\cdot)$$ 571.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{571}(2,\cdot)$$ 571.m 114 yes $$-1$$ $$1$$ $$e\left(\frac{11}{114}\right)$$ $$e\left(\frac{109}{114}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{28}{57}\right)$$ $$e\left(\frac{1}{19}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{11}{38}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{67}{114}\right)$$ $$e\left(\frac{14}{57}\right)$$
$$\chi_{571}(3,\cdot)$$ 571.p 570 yes $$-1$$ $$1$$ $$e\left(\frac{109}{114}\right)$$ $$e\left(\frac{1}{570}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{211}{285}\right)$$ $$e\left(\frac{91}{95}\right)$$ $$e\left(\frac{77}{190}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{1}{285}\right)$$ $$e\left(\frac{397}{570}\right)$$ $$e\left(\frac{134}{285}\right)$$
$$\chi_{571}(4,\cdot)$$ 571.k 57 yes $$1$$ $$1$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{22}{57}\right)$$ $$e\left(\frac{56}{57}\right)$$ $$e\left(\frac{2}{19}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{47}{57}\right)$$ $$e\left(\frac{10}{57}\right)$$ $$e\left(\frac{28}{57}\right)$$
$$\chi_{571}(5,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{28}{57}\right)$$ $$e\left(\frac{211}{285}\right)$$ $$e\left(\frac{56}{57}\right)$$ $$e\left(\frac{122}{285}\right)$$ $$e\left(\frac{22}{95}\right)$$ $$e\left(\frac{2}{95}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{137}{285}\right)$$ $$e\left(\frac{262}{285}\right)$$ $$e\left(\frac{118}{285}\right)$$
$$\chi_{571}(6,\cdot)$$ 571.l 95 yes $$1$$ $$1$$ $$e\left(\frac{1}{19}\right)$$ $$e\left(\frac{91}{95}\right)$$ $$e\left(\frac{2}{19}\right)$$ $$e\left(\frac{22}{95}\right)$$ $$e\left(\frac{1}{95}\right)$$ $$e\left(\frac{26}{95}\right)$$ $$e\left(\frac{3}{19}\right)$$ $$e\left(\frac{87}{95}\right)$$ $$e\left(\frac{27}{95}\right)$$ $$e\left(\frac{68}{95}\right)$$
$$\chi_{571}(7,\cdot)$$ 571.n 190 yes $$-1$$ $$1$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{77}{190}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{2}{95}\right)$$ $$e\left(\frac{26}{95}\right)$$ $$e\left(\frac{117}{190}\right)$$ $$e\left(\frac{23}{38}\right)$$ $$e\left(\frac{77}{95}\right)$$ $$e\left(\frac{169}{190}\right)$$ $$e\left(\frac{58}{95}\right)$$
$$\chi_{571}(8,\cdot)$$ 571.j 38 yes $$-1$$ $$1$$ $$e\left(\frac{11}{38}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{3}{19}\right)$$ $$e\left(\frac{23}{38}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{29}{38}\right)$$ $$e\left(\frac{14}{19}\right)$$
$$\chi_{571}(9,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{1}{285}\right)$$ $$e\left(\frac{47}{57}\right)$$ $$e\left(\frac{137}{285}\right)$$ $$e\left(\frac{87}{95}\right)$$ $$e\left(\frac{77}{95}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{2}{285}\right)$$ $$e\left(\frac{112}{285}\right)$$ $$e\left(\frac{268}{285}\right)$$
$$\chi_{571}(10,\cdot)$$ 571.p 570 yes $$-1$$ $$1$$ $$e\left(\frac{67}{114}\right)$$ $$e\left(\frac{397}{570}\right)$$ $$e\left(\frac{10}{57}\right)$$ $$e\left(\frac{262}{285}\right)$$ $$e\left(\frac{27}{95}\right)$$ $$e\left(\frac{169}{190}\right)$$ $$e\left(\frac{29}{38}\right)$$ $$e\left(\frac{112}{285}\right)$$ $$e\left(\frac{289}{570}\right)$$ $$e\left(\frac{188}{285}\right)$$
$$\chi_{571}(11,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{134}{285}\right)$$ $$e\left(\frac{28}{57}\right)$$ $$e\left(\frac{118}{285}\right)$$ $$e\left(\frac{68}{95}\right)$$ $$e\left(\frac{58}{95}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{268}{285}\right)$$ $$e\left(\frac{188}{285}\right)$$ $$e\left(\frac{2}{285}\right)$$
$$\chi_{571}(12,\cdot)$$ 571.p 570 yes $$-1$$ $$1$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{521}{570}\right)$$ $$e\left(\frac{17}{57}\right)$$ $$e\left(\frac{206}{285}\right)$$ $$e\left(\frac{6}{95}\right)$$ $$e\left(\frac{27}{190}\right)$$ $$e\left(\frac{17}{38}\right)$$ $$e\left(\frac{236}{285}\right)$$ $$e\left(\frac{497}{570}\right)$$ $$e\left(\frac{274}{285}\right)$$
$$\chi_{571}(13,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{32}{57}\right)$$ $$e\left(\frac{176}{285}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{172}{285}\right)$$ $$e\left(\frac{17}{95}\right)$$ $$e\left(\frac{62}{95}\right)$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{67}{285}\right)$$ $$e\left(\frac{47}{285}\right)$$ $$e\left(\frac{143}{285}\right)$$
$$\chi_{571}(14,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{103}{285}\right)$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{146}{285}\right)$$ $$e\left(\frac{31}{95}\right)$$ $$e\left(\frac{46}{95}\right)$$ $$e\left(\frac{17}{19}\right)$$ $$e\left(\frac{206}{285}\right)$$ $$e\left(\frac{136}{285}\right)$$ $$e\left(\frac{244}{285}\right)$$
$$\chi_{571}(15,\cdot)$$ 571.n 190 yes $$-1$$ $$1$$ $$e\left(\frac{17}{38}\right)$$ $$e\left(\frac{141}{190}\right)$$ $$e\left(\frac{17}{19}\right)$$ $$e\left(\frac{16}{95}\right)$$ $$e\left(\frac{18}{95}\right)$$ $$e\left(\frac{81}{190}\right)$$ $$e\left(\frac{13}{38}\right)$$ $$e\left(\frac{46}{95}\right)$$ $$e\left(\frac{117}{190}\right)$$ $$e\left(\frac{84}{95}\right)$$
$$\chi_{571}(16,\cdot)$$ 571.k 57 yes $$1$$ $$1$$ $$e\left(\frac{22}{57}\right)$$ $$e\left(\frac{47}{57}\right)$$ $$e\left(\frac{44}{57}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{3}{19}\right)$$ $$e\left(\frac{37}{57}\right)$$ $$e\left(\frac{20}{57}\right)$$ $$e\left(\frac{56}{57}\right)$$
$$\chi_{571}(17,\cdot)$$ 571.p 570 yes $$-1$$ $$1$$ $$e\left(\frac{101}{114}\right)$$ $$e\left(\frac{413}{570}\right)$$ $$e\left(\frac{44}{57}\right)$$ $$e\left(\frac{218}{285}\right)$$ $$e\left(\frac{58}{95}\right)$$ $$e\left(\frac{71}{190}\right)$$ $$e\left(\frac{25}{38}\right)$$ $$e\left(\frac{128}{285}\right)$$ $$e\left(\frac{371}{570}\right)$$ $$e\left(\frac{52}{285}\right)$$
$$\chi_{571}(18,\cdot)$$ 571.p 570 yes $$-1$$ $$1$$ $$e\left(\frac{1}{114}\right)$$ $$e\left(\frac{547}{570}\right)$$ $$e\left(\frac{1}{57}\right)$$ $$e\left(\frac{277}{285}\right)$$ $$e\left(\frac{92}{95}\right)$$ $$e\left(\frac{129}{190}\right)$$ $$e\left(\frac{1}{38}\right)$$ $$e\left(\frac{262}{285}\right)$$ $$e\left(\frac{559}{570}\right)$$ $$e\left(\frac{53}{285}\right)$$
$$\chi_{571}(19,\cdot)$$ 571.p 570 yes $$-1$$ $$1$$ $$e\left(\frac{109}{114}\right)$$ $$e\left(\frac{457}{570}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{97}{285}\right)$$ $$e\left(\frac{72}{95}\right)$$ $$e\left(\frac{39}{190}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{172}{285}\right)$$ $$e\left(\frac{169}{570}\right)$$ $$e\left(\frac{248}{285}\right)$$
$$\chi_{571}(20,\cdot)$$ 571.l 95 yes $$1$$ $$1$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{62}{95}\right)$$ $$e\left(\frac{7}{19}\right)$$ $$e\left(\frac{39}{95}\right)$$ $$e\left(\frac{32}{95}\right)$$ $$e\left(\frac{72}{95}\right)$$ $$e\left(\frac{1}{19}\right)$$ $$e\left(\frac{29}{95}\right)$$ $$e\left(\frac{9}{95}\right)$$ $$e\left(\frac{86}{95}\right)$$
$$\chi_{571}(21,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{47}{57}\right)$$ $$e\left(\frac{116}{285}\right)$$ $$e\left(\frac{37}{57}\right)$$ $$e\left(\frac{217}{285}\right)$$ $$e\left(\frac{22}{95}\right)$$ $$e\left(\frac{2}{95}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{232}{285}\right)$$ $$e\left(\frac{167}{285}\right)$$ $$e\left(\frac{23}{285}\right)$$
$$\chi_{571}(22,\cdot)$$ 571.n 190 yes $$-1$$ $$1$$ $$e\left(\frac{13}{38}\right)$$ $$e\left(\frac{81}{190}\right)$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{86}{95}\right)$$ $$e\left(\frac{73}{95}\right)$$ $$e\left(\frac{91}{190}\right)$$ $$e\left(\frac{1}{38}\right)$$ $$e\left(\frac{81}{95}\right)$$ $$e\left(\frac{47}{190}\right)$$ $$e\left(\frac{24}{95}\right)$$
$$\chi_{571}(23,\cdot)$$ 571.l 95 yes $$1$$ $$1$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{41}{95}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{12}{95}\right)$$ $$e\left(\frac{61}{95}\right)$$ $$e\left(\frac{66}{95}\right)$$ $$e\left(\frac{12}{19}\right)$$ $$e\left(\frac{82}{95}\right)$$ $$e\left(\frac{32}{95}\right)$$ $$e\left(\frac{63}{95}\right)$$
$$\chi_{571}(24,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{248}{285}\right)$$ $$e\left(\frac{28}{57}\right)$$ $$e\left(\frac{61}{285}\right)$$ $$e\left(\frac{11}{95}\right)$$ $$e\left(\frac{1}{95}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{211}{285}\right)$$ $$e\left(\frac{131}{285}\right)$$ $$e\left(\frac{59}{285}\right)$$
$$\chi_{571}(25,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{56}{57}\right)$$ $$e\left(\frac{137}{285}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{244}{285}\right)$$ $$e\left(\frac{44}{95}\right)$$ $$e\left(\frac{4}{95}\right)$$ $$e\left(\frac{18}{19}\right)$$ $$e\left(\frac{274}{285}\right)$$ $$e\left(\frac{239}{285}\right)$$ $$e\left(\frac{236}{285}\right)$$
$$\chi_{571}(26,\cdot)$$ 571.n 190 yes $$-1$$ $$1$$ $$e\left(\frac{25}{38}\right)$$ $$e\left(\frac{109}{190}\right)$$ $$e\left(\frac{6}{19}\right)$$ $$e\left(\frac{9}{95}\right)$$ $$e\left(\frac{22}{95}\right)$$ $$e\left(\frac{99}{190}\right)$$ $$e\left(\frac{37}{38}\right)$$ $$e\left(\frac{14}{95}\right)$$ $$e\left(\frac{143}{190}\right)$$ $$e\left(\frac{71}{95}\right)$$
$$\chi_{571}(27,\cdot)$$ 571.n 190 yes $$-1$$ $$1$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{1}{190}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{21}{95}\right)$$ $$e\left(\frac{83}{95}\right)$$ $$e\left(\frac{41}{190}\right)$$ $$e\left(\frac{23}{38}\right)$$ $$e\left(\frac{1}{95}\right)$$ $$e\left(\frac{17}{190}\right)$$ $$e\left(\frac{39}{95}\right)$$
$$\chi_{571}(28,\cdot)$$ 571.p 570 yes $$-1$$ $$1$$ $$e\left(\frac{7}{114}\right)$$ $$e\left(\frac{181}{570}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{1}{285}\right)$$ $$e\left(\frac{36}{95}\right)$$ $$e\left(\frac{67}{190}\right)$$ $$e\left(\frac{7}{38}\right)$$ $$e\left(\frac{181}{285}\right)$$ $$e\left(\frac{37}{570}\right)$$ $$e\left(\frac{29}{285}\right)$$
$$\chi_{571}(29,\cdot)$$ 571.k 57 yes $$1$$ $$1$$ $$e\left(\frac{13}{57}\right)$$ $$e\left(\frac{20}{57}\right)$$ $$e\left(\frac{26}{57}\right)$$ $$e\left(\frac{4}{57}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{1}{19}\right)$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{40}{57}\right)$$ $$e\left(\frac{17}{57}\right)$$ $$e\left(\frac{2}{57}\right)$$
$$\chi_{571}(30,\cdot)$$ 571.o 285 yes $$1$$ $$1$$ $$e\left(\frac{31}{57}\right)$$ $$e\left(\frac{199}{285}\right)$$ $$e\left(\frac{5}{57}\right)$$ $$e\left(\frac{188}{285}\right)$$ $$e\left(\frac{23}{95}\right)$$ $$e\left(\frac{28}{95}\right)$$ $$e\left(\frac{12}{19}\right)$$ $$e\left(\frac{113}{285}\right)$$ $$e\left(\frac{58}{285}\right)$$ $$e\left(\frac{37}{285}\right)$$
$$\chi_{571}(31,\cdot)$$ 571.h 19 yes $$1$$ $$1$$ $$e\left(\frac{1}{19}\right)$$ $$e\left(\frac{3}{19}\right)$$ $$e\left(\frac{2}{19}\right)$$ $$e\left(\frac{12}{19}\right)$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{3}{19}\right)$$ $$e\left(\frac{6}{19}\right)$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{6}{19}\right)$$
$$\chi_{571}(32,\cdot)$$ 571.m 114 yes $$-1$$ $$1$$ $$e\left(\frac{55}{114}\right)$$ $$e\left(\frac{89}{114}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{26}{57}\right)$$ $$e\left(\frac{5}{19}\right)$$ $$e\left(\frac{13}{38}\right)$$ $$e\left(\frac{17}{38}\right)$$ $$e\left(\frac{32}{57}\right)$$ $$e\left(\frac{107}{114}\right)$$ $$e\left(\frac{13}{57}\right)$$
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