# Properties

 Modulus $723$ Structure $$C_{240}\times C_{2}$$ Order $480$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(723)

pari: g = idealstar(,723,2)

## Character group

 sage: G.order()  pari: g.no Order = 480 sage: H.invariants()  pari: g.cyc Structure = $$C_{240}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{723}(242,\cdot)$, $\chi_{723}(7,\cdot)$

## First 32 of 480 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{723}(1,\cdot)$$ 723.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{723}(2,\cdot)$$ 723.x 24 yes $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{19}{24}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{723}(4,\cdot)$$ 723.r 12 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{723}(5,\cdot)$$ 723.bc 40 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$1$$
$$\chi_{723}(7,\cdot)$$ 723.bm 240 no $$-1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{1}{240}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{47}{240}\right)$$ $$e\left(\frac{191}{240}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{723}(8,\cdot)$$ 723.m 8 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{723}(10,\cdot)$$ 723.z 30 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{723}(11,\cdot)$$ 723.be 48 yes $$1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{723}(13,\cdot)$$ 723.bm 240 no $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{47}{240}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{49}{240}\right)$$ $$e\left(\frac{97}{240}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{723}(14,\cdot)$$ 723.bn 240 yes $$1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{191}{240}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{97}{240}\right)$$ $$e\left(\frac{121}{240}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{723}(16,\cdot)$$ 723.i 6 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{723}(17,\cdot)$$ 723.bj 80 yes $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{37}{80}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{59}{80}\right)$$ $$e\left(\frac{67}{80}\right)$$ $$-1$$
$$\chi_{723}(19,\cdot)$$ 723.bf 48 no $$-1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{723}(20,\cdot)$$ 723.bl 120 yes $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$-i$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{53}{120}\right)$$ $$e\left(\frac{89}{120}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{723}(22,\cdot)$$ 723.bf 48 no $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{723}(23,\cdot)$$ 723.bj 80 yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{19}{80}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{80}\right)$$ $$e\left(\frac{69}{80}\right)$$ $$-1$$
$$\chi_{723}(25,\cdot)$$ 723.v 20 no $$1$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$-i$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$1$$
$$\chi_{723}(26,\cdot)$$ 723.bj 80 yes $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{79}{80}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{33}{80}\right)$$ $$e\left(\frac{9}{80}\right)$$ $$-1$$
$$\chi_{723}(28,\cdot)$$ 723.bi 80 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{47}{80}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{49}{80}\right)$$ $$e\left(\frac{17}{80}\right)$$ $$-1$$
$$\chi_{723}(29,\cdot)$$ 723.bl 120 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{77}{120}\right)$$ $$i$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{723}(31,\cdot)$$ 723.bm 240 no $$-1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{151}{240}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{137}{240}\right)$$ $$e\left(\frac{41}{240}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{723}(32,\cdot)$$ 723.x 24 yes $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{23}{24}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{723}(34,\cdot)$$ 723.bm 240 no $$-1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{61}{240}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{227}{240}\right)$$ $$e\left(\frac{131}{240}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{723}(35,\cdot)$$ 723.bn 240 yes $$1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{139}{240}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{53}{240}\right)$$ $$e\left(\frac{29}{240}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{723}(37,\cdot)$$ 723.bm 240 no $$-1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{73}{240}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{71}{240}\right)$$ $$e\left(\frac{23}{240}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{723}(38,\cdot)$$ 723.be 48 yes $$1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{723}(40,\cdot)$$ 723.v 20 no $$1$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$-i$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$1$$
$$\chi_{723}(41,\cdot)$$ 723.bc 40 yes $$-1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$-i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$1$$
$$\chi_{723}(43,\cdot)$$ 723.bi 80 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{73}{80}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{71}{80}\right)$$ $$e\left(\frac{23}{80}\right)$$ $$-1$$
$$\chi_{723}(44,\cdot)$$ 723.t 16 yes $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-1$$
$$\chi_{723}(46,\cdot)$$ 723.bm 240 no $$-1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{7}{240}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{89}{240}\right)$$ $$e\left(\frac{137}{240}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{723}(47,\cdot)$$ 723.bc 40 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$1$$