# Properties

 Modulus $2009$ Structure $$C_{840}\times C_{2}$$ Order $1680$

# Learn more about

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(2009)

pari: g = idealstar(,2009,2)

## Character group

 sage: G.order()  pari: g.no Order = 1680 sage: H.invariants()  pari: g.cyc Structure = $$C_{840}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{2009}(493,\cdot)$, $\chi_{2009}(785,\cdot)$

## First 32 of 1680 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$
$$\chi_{2009}(1,\cdot)$$ 2009.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{2009}(2,\cdot)$$ 2009.cj 420 yes $$1$$ $$1$$ $$e\left(\frac{209}{210}\right)$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{53}{210}\right)$$ $$e\left(\frac{51}{140}\right)$$ $$e\left(\frac{69}{70}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{26}{105}\right)$$ $$e\left(\frac{299}{420}\right)$$ $$e\left(\frac{151}{420}\right)$$
$$\chi_{2009}(3,\cdot)$$ 2009.cb 168 yes $$1$$ $$1$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{109}{168}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{1}{56}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{168}\right)$$ $$e\left(\frac{65}{168}\right)$$
$$\chi_{2009}(4,\cdot)$$ 2009.cd 210 yes $$1$$ $$1$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{51}{70}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{89}{210}\right)$$ $$e\left(\frac{151}{210}\right)$$
$$\chi_{2009}(5,\cdot)$$ 2009.ci 420 yes $$-1$$ $$1$$ $$e\left(\frac{53}{210}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{27}{140}\right)$$ $$e\left(\frac{53}{70}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{79}{210}\right)$$ $$e\left(\frac{113}{420}\right)$$ $$e\left(\frac{187}{420}\right)$$
$$\chi_{2009}(6,\cdot)$$ 2009.cg 280 yes $$1$$ $$1$$ $$e\left(\frac{51}{140}\right)$$ $$e\left(\frac{1}{56}\right)$$ $$e\left(\frac{51}{70}\right)$$ $$e\left(\frac{27}{140}\right)$$ $$e\left(\frac{107}{280}\right)$$ $$e\left(\frac{13}{140}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{39}{70}\right)$$ $$e\left(\frac{221}{280}\right)$$ $$e\left(\frac{209}{280}\right)$$
$$\chi_{2009}(8,\cdot)$$ 2009.bz 140 yes $$1$$ $$1$$ $$e\left(\frac{69}{70}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{53}{70}\right)$$ $$e\left(\frac{13}{140}\right)$$ $$e\left(\frac{67}{70}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{19}{140}\right)$$ $$e\left(\frac{11}{140}\right)$$
$$\chi_{2009}(9,\cdot)$$ 2009.bu 84 yes $$1$$ $$1$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{65}{84}\right)$$
$$\chi_{2009}(10,\cdot)$$ 2009.ce 210 yes $$-1$$ $$1$$ $$e\left(\frac{26}{105}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{52}{105}\right)$$ $$e\left(\frac{79}{210}\right)$$ $$e\left(\frac{39}{70}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{131}{210}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{169}{210}\right)$$
$$\chi_{2009}(11,\cdot)$$ 2009.ck 840 yes $$-1$$ $$1$$ $$e\left(\frac{299}{420}\right)$$ $$e\left(\frac{13}{168}\right)$$ $$e\left(\frac{89}{210}\right)$$ $$e\left(\frac{113}{420}\right)$$ $$e\left(\frac{221}{280}\right)$$ $$e\left(\frac{19}{140}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{269}{840}\right)$$ $$e\left(\frac{421}{840}\right)$$
$$\chi_{2009}(12,\cdot)$$ 2009.cl 840 yes $$1$$ $$1$$ $$e\left(\frac{151}{420}\right)$$ $$e\left(\frac{65}{168}\right)$$ $$e\left(\frac{151}{210}\right)$$ $$e\left(\frac{187}{420}\right)$$ $$e\left(\frac{209}{280}\right)$$ $$e\left(\frac{11}{140}\right)$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{169}{210}\right)$$ $$e\left(\frac{421}{840}\right)$$ $$e\left(\frac{89}{840}\right)$$
$$\chi_{2009}(13,\cdot)$$ 2009.cg 280 yes $$1$$ $$1$$ $$e\left(\frac{81}{140}\right)$$ $$e\left(\frac{23}{56}\right)$$ $$e\left(\frac{11}{70}\right)$$ $$e\left(\frac{117}{140}\right)$$ $$e\left(\frac{277}{280}\right)$$ $$e\left(\frac{103}{140}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{29}{70}\right)$$ $$e\left(\frac{211}{280}\right)$$ $$e\left(\frac{159}{280}\right)$$
$$\chi_{2009}(15,\cdot)$$ 2009.ch 280 yes $$-1$$ $$1$$ $$e\left(\frac{87}{140}\right)$$ $$e\left(\frac{33}{56}\right)$$ $$e\left(\frac{17}{70}\right)$$ $$e\left(\frac{9}{140}\right)$$ $$e\left(\frac{59}{280}\right)$$ $$e\left(\frac{121}{140}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{97}{280}\right)$$ $$e\left(\frac{233}{280}\right)$$
$$\chi_{2009}(16,\cdot)$$ 2009.bw 105 yes $$1$$ $$1$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{46}{105}\right)$$
$$\chi_{2009}(17,\cdot)$$ 2009.cl 840 yes $$1$$ $$1$$ $$e\left(\frac{389}{420}\right)$$ $$e\left(\frac{163}{168}\right)$$ $$e\left(\frac{179}{210}\right)$$ $$e\left(\frac{173}{420}\right)$$ $$e\left(\frac{251}{280}\right)$$ $$e\left(\frac{109}{140}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{71}{210}\right)$$ $$e\left(\frac{239}{840}\right)$$ $$e\left(\frac{691}{840}\right)$$
$$\chi_{2009}(18,\cdot)$$ 2009.w 15 no $$1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$
$$\chi_{2009}(19,\cdot)$$ 2009.bx 120 no $$1$$ $$1$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{120}\right)$$ $$e\left(\frac{29}{120}\right)$$
$$\chi_{2009}(20,\cdot)$$ 2009.ca 140 yes $$-1$$ $$1$$ $$e\left(\frac{17}{70}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{129}{140}\right)$$ $$e\left(\frac{51}{70}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{61}{70}\right)$$ $$e\left(\frac{97}{140}\right)$$ $$e\left(\frac{23}{140}\right)$$
$$\chi_{2009}(22,\cdot)$$ 2009.ch 280 yes $$-1$$ $$1$$ $$e\left(\frac{99}{140}\right)$$ $$e\left(\frac{25}{56}\right)$$ $$e\left(\frac{29}{70}\right)$$ $$e\left(\frac{73}{140}\right)$$ $$e\left(\frac{43}{280}\right)$$ $$e\left(\frac{17}{140}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{9}{280}\right)$$ $$e\left(\frac{241}{280}\right)$$
$$\chi_{2009}(23,\cdot)$$ 2009.cd 210 yes $$1$$ $$1$$ $$e\left(\frac{97}{105}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{89}{105}\right)$$ $$e\left(\frac{4}{105}\right)$$ $$e\left(\frac{23}{70}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{101}{105}\right)$$ $$e\left(\frac{187}{210}\right)$$ $$e\left(\frac{53}{210}\right)$$
$$\chi_{2009}(24,\cdot)$$ 2009.cl 840 yes $$1$$ $$1$$ $$e\left(\frac{149}{420}\right)$$ $$e\left(\frac{127}{168}\right)$$ $$e\left(\frac{149}{210}\right)$$ $$e\left(\frac{293}{420}\right)$$ $$e\left(\frac{31}{280}\right)$$ $$e\left(\frac{9}{140}\right)$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{11}{210}\right)$$ $$e\left(\frac{179}{840}\right)$$ $$e\left(\frac{391}{840}\right)$$
$$\chi_{2009}(25,\cdot)$$ 2009.cd 210 yes $$1$$ $$1$$ $$e\left(\frac{53}{105}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{105}\right)$$ $$e\left(\frac{26}{105}\right)$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{113}{210}\right)$$ $$e\left(\frac{187}{210}\right)$$
$$\chi_{2009}(26,\cdot)$$ 2009.cl 840 yes $$1$$ $$1$$ $$e\left(\frac{241}{420}\right)$$ $$e\left(\frac{131}{168}\right)$$ $$e\left(\frac{31}{210}\right)$$ $$e\left(\frac{37}{420}\right)$$ $$e\left(\frac{99}{280}\right)$$ $$e\left(\frac{101}{140}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{139}{210}\right)$$ $$e\left(\frac{391}{840}\right)$$ $$e\left(\frac{779}{840}\right)$$
$$\chi_{2009}(27,\cdot)$$ 2009.bo 56 yes $$1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{53}{56}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{3}{56}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{13}{56}\right)$$ $$e\left(\frac{9}{56}\right)$$
$$\chi_{2009}(29,\cdot)$$ 2009.ch 280 yes $$-1$$ $$1$$ $$e\left(\frac{97}{140}\right)$$ $$e\left(\frac{3}{56}\right)$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{39}{140}\right)$$ $$e\left(\frac{209}{280}\right)$$ $$e\left(\frac{11}{140}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{187}{280}\right)$$ $$e\left(\frac{123}{280}\right)$$
$$\chi_{2009}(30,\cdot)$$ 2009.by 120 no $$-1$$ $$1$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{23}{120}\right)$$
$$\chi_{2009}(31,\cdot)$$ 2009.be 30 no $$-1$$ $$1$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$
$$\chi_{2009}(32,\cdot)$$ 2009.bu 84 yes $$1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{67}{84}\right)$$
$$\chi_{2009}(33,\cdot)$$ 2009.ci 420 yes $$-1$$ $$1$$ $$e\left(\frac{17}{210}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{113}{140}\right)$$ $$e\left(\frac{17}{70}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{61}{210}\right)$$ $$e\left(\frac{167}{420}\right)$$ $$e\left(\frac{373}{420}\right)$$
$$\chi_{2009}(34,\cdot)$$ 2009.cg 280 yes $$1$$ $$1$$ $$e\left(\frac{129}{140}\right)$$ $$e\left(\frac{19}{56}\right)$$ $$e\left(\frac{59}{70}\right)$$ $$e\left(\frac{93}{140}\right)$$ $$e\left(\frac{73}{280}\right)$$ $$e\left(\frac{107}{140}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{41}{70}\right)$$ $$e\left(\frac{279}{280}\right)$$ $$e\left(\frac{51}{280}\right)$$
$$\chi_{2009}(36,\cdot)$$ 2009.bz 140 yes $$1$$ $$1$$ $$e\left(\frac{51}{70}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{107}{140}\right)$$ $$e\left(\frac{13}{70}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{81}{140}\right)$$ $$e\left(\frac{69}{140}\right)$$
$$\chi_{2009}(37,\cdot)$$ 2009.bw 105 yes $$1$$ $$1$$ $$e\left(\frac{64}{105}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{73}{105}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{32}{105}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{103}{105}\right)$$