# Properties

 Modulus $547$ Structure $$C_{546}$$ Order $546$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(547)

pari: g = idealstar(,547,2)

## Character group

 sage: G.order()  pari: g.no Order = 546 sage: H.invariants()  pari: g.cyc Structure = $$C_{546}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{547}(2,\cdot)$

## First 32 of 546 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{547}(1,\cdot)$$ 547.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{547}(2,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{1}{546}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{273}\right)$$ $$e\left(\frac{179}{546}\right)$$ $$e\left(\frac{215}{273}\right)$$ $$e\left(\frac{103}{546}\right)$$ $$e\left(\frac{1}{182}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{30}{91}\right)$$ $$e\left(\frac{5}{39}\right)$$
$$\chi_{547}(3,\cdot)$$ 547.g 14 yes $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$1$$
$$\chi_{547}(4,\cdot)$$ 547.o 273 yes $$1$$ $$1$$ $$e\left(\frac{1}{273}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{273}\right)$$ $$e\left(\frac{179}{273}\right)$$ $$e\left(\frac{157}{273}\right)$$ $$e\left(\frac{103}{273}\right)$$ $$e\left(\frac{1}{91}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{60}{91}\right)$$ $$e\left(\frac{10}{39}\right)$$
$$\chi_{547}(5,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{179}{546}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{179}{273}\right)$$ $$e\left(\frac{373}{546}\right)$$ $$e\left(\frac{265}{273}\right)$$ $$e\left(\frac{419}{546}\right)$$ $$e\left(\frac{179}{182}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{91}\right)$$ $$e\left(\frac{37}{39}\right)$$
$$\chi_{547}(6,\cdot)$$ 547.o 273 yes $$1$$ $$1$$ $$e\left(\frac{215}{273}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{157}{273}\right)$$ $$e\left(\frac{265}{273}\right)$$ $$e\left(\frac{176}{273}\right)$$ $$e\left(\frac{32}{273}\right)$$ $$e\left(\frac{33}{91}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{69}{91}\right)$$ $$e\left(\frac{5}{39}\right)$$
$$\chi_{547}(7,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{103}{546}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{103}{273}\right)$$ $$e\left(\frac{419}{546}\right)$$ $$e\left(\frac{32}{273}\right)$$ $$e\left(\frac{235}{546}\right)$$ $$e\left(\frac{103}{182}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{87}{91}\right)$$ $$e\left(\frac{8}{39}\right)$$
$$\chi_{547}(8,\cdot)$$ 547.n 182 yes $$-1$$ $$1$$ $$e\left(\frac{1}{182}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{91}\right)$$ $$e\left(\frac{179}{182}\right)$$ $$e\left(\frac{33}{91}\right)$$ $$e\left(\frac{103}{182}\right)$$ $$e\left(\frac{3}{182}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{90}{91}\right)$$ $$e\left(\frac{5}{13}\right)$$
$$\chi_{547}(9,\cdot)$$ 547.e 7 yes $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$1$$
$$\chi_{547}(10,\cdot)$$ 547.m 91 yes $$1$$ $$1$$ $$e\left(\frac{30}{91}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{60}{91}\right)$$ $$e\left(\frac{1}{91}\right)$$ $$e\left(\frac{69}{91}\right)$$ $$e\left(\frac{87}{91}\right)$$ $$e\left(\frac{90}{91}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{31}{91}\right)$$ $$e\left(\frac{1}{13}\right)$$
$$\chi_{547}(11,\cdot)$$ 547.j 39 yes $$1$$ $$1$$ $$e\left(\frac{5}{39}\right)$$ $$1$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$1$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{38}{39}\right)$$
$$\chi_{547}(12,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{431}{546}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{158}{273}\right)$$ $$e\left(\frac{163}{546}\right)$$ $$e\left(\frac{118}{273}\right)$$ $$e\left(\frac{167}{546}\right)$$ $$e\left(\frac{67}{182}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{8}{91}\right)$$ $$e\left(\frac{10}{39}\right)$$
$$\chi_{547}(13,\cdot)$$ 547.h 21 yes $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{547}(14,\cdot)$$ 547.h 21 yes $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{547}(15,\cdot)$$ 547.o 273 yes $$1$$ $$1$$ $$e\left(\frac{31}{273}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{62}{273}\right)$$ $$e\left(\frac{89}{273}\right)$$ $$e\left(\frac{226}{273}\right)$$ $$e\left(\frac{190}{273}\right)$$ $$e\left(\frac{31}{91}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{40}{91}\right)$$ $$e\left(\frac{37}{39}\right)$$
$$\chi_{547}(16,\cdot)$$ 547.o 273 yes $$1$$ $$1$$ $$e\left(\frac{2}{273}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{273}\right)$$ $$e\left(\frac{85}{273}\right)$$ $$e\left(\frac{41}{273}\right)$$ $$e\left(\frac{206}{273}\right)$$ $$e\left(\frac{2}{91}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{29}{91}\right)$$ $$e\left(\frac{20}{39}\right)$$
$$\chi_{547}(17,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{151}{546}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{151}{273}\right)$$ $$e\left(\frac{275}{546}\right)$$ $$e\left(\frac{251}{273}\right)$$ $$e\left(\frac{265}{546}\right)$$ $$e\left(\frac{151}{182}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{71}{91}\right)$$ $$e\left(\frac{14}{39}\right)$$
$$\chi_{547}(18,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{313}{546}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{40}{273}\right)$$ $$e\left(\frac{335}{546}\right)$$ $$e\left(\frac{137}{273}\right)$$ $$e\left(\frac{25}{546}\right)$$ $$e\left(\frac{131}{182}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{17}{91}\right)$$ $$e\left(\frac{5}{39}\right)$$
$$\chi_{547}(19,\cdot)$$ 547.o 273 yes $$1$$ $$1$$ $$e\left(\frac{115}{273}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{230}{273}\right)$$ $$e\left(\frac{110}{273}\right)$$ $$e\left(\frac{37}{273}\right)$$ $$e\left(\frac{106}{273}\right)$$ $$e\left(\frac{24}{91}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{75}{91}\right)$$ $$e\left(\frac{19}{39}\right)$$
$$\chi_{547}(20,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{181}{546}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{181}{273}\right)$$ $$e\left(\frac{185}{546}\right)$$ $$e\left(\frac{149}{273}\right)$$ $$e\left(\frac{79}{546}\right)$$ $$e\left(\frac{181}{182}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{61}{91}\right)$$ $$e\left(\frac{8}{39}\right)$$
$$\chi_{547}(21,\cdot)$$ 547.j 39 yes $$1$$ $$1$$ $$e\left(\frac{38}{39}\right)$$ $$1$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$1$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{8}{39}\right)$$
$$\chi_{547}(22,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{71}{546}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{71}{273}\right)$$ $$e\left(\frac{151}{546}\right)$$ $$e\left(\frac{250}{273}\right)$$ $$e\left(\frac{215}{546}\right)$$ $$e\left(\frac{71}{182}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{37}{91}\right)$$ $$e\left(\frac{4}{39}\right)$$
$$\chi_{547}(23,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{293}{546}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{20}{273}\right)$$ $$e\left(\frac{31}{546}\right)$$ $$e\left(\frac{205}{273}\right)$$ $$e\left(\frac{149}{546}\right)$$ $$e\left(\frac{111}{182}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{54}{91}\right)$$ $$e\left(\frac{22}{39}\right)$$
$$\chi_{547}(24,\cdot)$$ 547.m 91 yes $$1$$ $$1$$ $$e\left(\frac{72}{91}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{53}{91}\right)$$ $$e\left(\frac{57}{91}\right)$$ $$e\left(\frac{20}{91}\right)$$ $$e\left(\frac{45}{91}\right)$$ $$e\left(\frac{34}{91}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{38}{91}\right)$$ $$e\left(\frac{5}{13}\right)$$
$$\chi_{547}(25,\cdot)$$ 547.o 273 yes $$1$$ $$1$$ $$e\left(\frac{179}{273}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{85}{273}\right)$$ $$e\left(\frac{100}{273}\right)$$ $$e\left(\frac{257}{273}\right)$$ $$e\left(\frac{146}{273}\right)$$ $$e\left(\frac{88}{91}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{91}\right)$$ $$e\left(\frac{35}{39}\right)$$
$$\chi_{547}(26,\cdot)$$ 547.l 78 yes $$-1$$ $$1$$ $$e\left(\frac{41}{78}\right)$$ $$-1$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$1$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{31}{39}\right)$$
$$\chi_{547}(27,\cdot)$$ 547.g 14 yes $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$
$$\chi_{547}(28,\cdot)$$ 547.i 26 yes $$-1$$ $$1$$ $$e\left(\frac{5}{26}\right)$$ $$-1$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$1$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$
$$\chi_{547}(29,\cdot)$$ 547.m 91 yes $$1$$ $$1$$ $$e\left(\frac{53}{91}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{15}{91}\right)$$ $$e\left(\frac{23}{91}\right)$$ $$e\left(\frac{40}{91}\right)$$ $$e\left(\frac{90}{91}\right)$$ $$e\left(\frac{68}{91}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{76}{91}\right)$$ $$e\left(\frac{10}{13}\right)$$
$$\chi_{547}(30,\cdot)$$ 547.i 26 yes $$-1$$ $$1$$ $$e\left(\frac{3}{26}\right)$$ $$-1$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$1$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$
$$\chi_{547}(31,\cdot)$$ 547.n 182 yes $$-1$$ $$1$$ $$e\left(\frac{101}{182}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{10}{91}\right)$$ $$e\left(\frac{61}{182}\right)$$ $$e\left(\frac{57}{91}\right)$$ $$e\left(\frac{29}{182}\right)$$ $$e\left(\frac{121}{182}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{81}{91}\right)$$ $$e\left(\frac{11}{13}\right)$$
$$\chi_{547}(32,\cdot)$$ 547.p 546 yes $$-1$$ $$1$$ $$e\left(\frac{5}{546}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{273}\right)$$ $$e\left(\frac{349}{546}\right)$$ $$e\left(\frac{256}{273}\right)$$ $$e\left(\frac{515}{546}\right)$$ $$e\left(\frac{5}{182}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{59}{91}\right)$$ $$e\left(\frac{25}{39}\right)$$