# Properties

 Modulus $1225$ Structure $$C_{420}\times C_{2}$$ Order $840$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(1225)

pari: g = idealstar(,1225,2)

## Character group

 sage: G.order()  pari: g.no Order = 840 sage: H.invariants()  pari: g.cyc Structure = $$C_{420}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1225}(1177,\cdot)$, $\chi_{1225}(101,\cdot)$

## First 32 of 840 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$16$$
$$\chi_{1225}(1,\cdot)$$ 1225.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1225}(2,\cdot)$$ 1225.bv 420 yes $$-1$$ $$1$$ $$e\left(\frac{61}{420}\right)$$ $$e\left(\frac{407}{420}\right)$$ $$e\left(\frac{61}{210}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{61}{140}\right)$$ $$e\left(\frac{197}{210}\right)$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{109}{420}\right)$$ $$e\left(\frac{53}{140}\right)$$ $$e\left(\frac{61}{105}\right)$$
$$\chi_{1225}(3,\cdot)$$ 1225.bu 420 yes $$1$$ $$1$$ $$e\left(\frac{407}{420}\right)$$ $$e\left(\frac{199}{420}\right)$$ $$e\left(\frac{197}{210}\right)$$ $$e\left(\frac{31}{70}\right)$$ $$e\left(\frac{127}{140}\right)$$ $$e\left(\frac{199}{210}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{173}{420}\right)$$ $$e\left(\frac{61}{140}\right)$$ $$e\left(\frac{92}{105}\right)$$
$$\chi_{1225}(4,\cdot)$$ 1225.bt 210 yes $$1$$ $$1$$ $$e\left(\frac{61}{210}\right)$$ $$e\left(\frac{197}{210}\right)$$ $$e\left(\frac{61}{105}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{61}{70}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{109}{210}\right)$$ $$e\left(\frac{53}{70}\right)$$ $$e\left(\frac{17}{105}\right)$$
$$\chi_{1225}(6,\cdot)$$ 1225.bl 70 yes $$-1$$ $$1$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{31}{70}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{39}{70}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{47}{70}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{16}{35}\right)$$
$$\chi_{1225}(8,\cdot)$$ 1225.bq 140 yes $$-1$$ $$1$$ $$e\left(\frac{61}{140}\right)$$ $$e\left(\frac{127}{140}\right)$$ $$e\left(\frac{61}{70}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{43}{140}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{109}{140}\right)$$ $$e\left(\frac{19}{140}\right)$$ $$e\left(\frac{26}{35}\right)$$
$$\chi_{1225}(9,\cdot)$$ 1225.bt 210 yes $$1$$ $$1$$ $$e\left(\frac{197}{210}\right)$$ $$e\left(\frac{199}{210}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{173}{210}\right)$$ $$e\left(\frac{61}{70}\right)$$ $$e\left(\frac{79}{105}\right)$$
$$\chi_{1225}(11,\cdot)$$ 1225.bo 105 yes $$1$$ $$1$$ $$e\left(\frac{59}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{13}{105}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{11}{105}\right)$$ $$e\left(\frac{94}{105}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{26}{105}\right)$$
$$\chi_{1225}(12,\cdot)$$ 1225.bu 420 yes $$1$$ $$1$$ $$e\left(\frac{109}{420}\right)$$ $$e\left(\frac{173}{420}\right)$$ $$e\left(\frac{109}{210}\right)$$ $$e\left(\frac{47}{70}\right)$$ $$e\left(\frac{109}{140}\right)$$ $$e\left(\frac{173}{210}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{391}{420}\right)$$ $$e\left(\frac{27}{140}\right)$$ $$e\left(\frac{4}{105}\right)$$
$$\chi_{1225}(13,\cdot)$$ 1225.bp 140 yes $$1$$ $$1$$ $$e\left(\frac{53}{140}\right)$$ $$e\left(\frac{61}{140}\right)$$ $$e\left(\frac{53}{70}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{19}{140}\right)$$ $$e\left(\frac{61}{70}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{27}{140}\right)$$ $$e\left(\frac{137}{140}\right)$$ $$e\left(\frac{18}{35}\right)$$
$$\chi_{1225}(16,\cdot)$$ 1225.bo 105 yes $$1$$ $$1$$ $$e\left(\frac{61}{105}\right)$$ $$e\left(\frac{92}{105}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{79}{105}\right)$$ $$e\left(\frac{26}{105}\right)$$ $$e\left(\frac{4}{105}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{34}{105}\right)$$
$$\chi_{1225}(17,\cdot)$$ 1225.bu 420 yes $$1$$ $$1$$ $$e\left(\frac{53}{420}\right)$$ $$e\left(\frac{61}{420}\right)$$ $$e\left(\frac{53}{210}\right)$$ $$e\left(\frac{19}{70}\right)$$ $$e\left(\frac{53}{140}\right)$$ $$e\left(\frac{61}{210}\right)$$ $$e\left(\frac{22}{105}\right)$$ $$e\left(\frac{167}{420}\right)$$ $$e\left(\frac{139}{140}\right)$$ $$e\left(\frac{53}{105}\right)$$
$$\chi_{1225}(18,\cdot)$$ 1225.q 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1225}(19,\cdot)$$ 1225.bb 30 no $$-1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$
$$\chi_{1225}(22,\cdot)$$ 1225.bq 140 yes $$-1$$ $$1$$ $$e\left(\frac{99}{140}\right)$$ $$e\left(\frac{73}{140}\right)$$ $$e\left(\frac{29}{70}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{17}{140}\right)$$ $$e\left(\frac{3}{70}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{131}{140}\right)$$ $$e\left(\frac{1}{140}\right)$$ $$e\left(\frac{29}{35}\right)$$
$$\chi_{1225}(23,\cdot)$$ 1225.bv 420 yes $$-1$$ $$1$$ $$e\left(\frac{31}{420}\right)$$ $$e\left(\frac{317}{420}\right)$$ $$e\left(\frac{31}{210}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{31}{140}\right)$$ $$e\left(\frac{107}{210}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{379}{420}\right)$$ $$e\left(\frac{43}{140}\right)$$ $$e\left(\frac{31}{105}\right)$$
$$\chi_{1225}(24,\cdot)$$ 1225.bf 42 no $$-1$$ $$1$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{1225}(26,\cdot)$$ 1225.bg 42 no $$-1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{21}\right)$$
$$\chi_{1225}(27,\cdot)$$ 1225.bp 140 yes $$1$$ $$1$$ $$e\left(\frac{127}{140}\right)$$ $$e\left(\frac{59}{140}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{23}{70}\right)$$ $$e\left(\frac{101}{140}\right)$$ $$e\left(\frac{59}{70}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{33}{140}\right)$$ $$e\left(\frac{43}{140}\right)$$ $$e\left(\frac{22}{35}\right)$$
$$\chi_{1225}(29,\cdot)$$ 1225.bj 70 yes $$1$$ $$1$$ $$e\left(\frac{17}{70}\right)$$ $$e\left(\frac{9}{70}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{51}{70}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{26}{35}\right)$$ $$e\left(\frac{43}{70}\right)$$ $$e\left(\frac{3}{70}\right)$$ $$e\left(\frac{34}{35}\right)$$
$$\chi_{1225}(31,\cdot)$$ 1225.bc 30 no $$-1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{14}{15}\right)$$
$$\chi_{1225}(32,\cdot)$$ 1225.bm 84 no $$-1$$ $$1$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{19}{21}\right)$$
$$\chi_{1225}(33,\cdot)$$ 1225.bu 420 yes $$1$$ $$1$$ $$e\left(\frac{223}{420}\right)$$ $$e\left(\frac{11}{420}\right)$$ $$e\left(\frac{13}{210}\right)$$ $$e\left(\frac{39}{70}\right)$$ $$e\left(\frac{83}{140}\right)$$ $$e\left(\frac{11}{210}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{37}{420}\right)$$ $$e\left(\frac{9}{140}\right)$$ $$e\left(\frac{13}{105}\right)$$
$$\chi_{1225}(34,\cdot)$$ 1225.bk 70 yes $$-1$$ $$1$$ $$e\left(\frac{19}{70}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{3}{35}\right)$$
$$\chi_{1225}(36,\cdot)$$ 1225.bd 35 yes $$1$$ $$1$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{32}{35}\right)$$
$$\chi_{1225}(37,\cdot)$$ 1225.bv 420 yes $$-1$$ $$1$$ $$e\left(\frac{109}{420}\right)$$ $$e\left(\frac{383}{420}\right)$$ $$e\left(\frac{109}{210}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{109}{140}\right)$$ $$e\left(\frac{173}{210}\right)$$ $$e\left(\frac{71}{105}\right)$$ $$e\left(\frac{181}{420}\right)$$ $$e\left(\frac{97}{140}\right)$$ $$e\left(\frac{4}{105}\right)$$
$$\chi_{1225}(38,\cdot)$$ 1225.bu 420 yes $$1$$ $$1$$ $$e\left(\frac{299}{420}\right)$$ $$e\left(\frac{43}{420}\right)$$ $$e\left(\frac{89}{210}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{19}{140}\right)$$ $$e\left(\frac{43}{210}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{221}{420}\right)$$ $$e\left(\frac{137}{140}\right)$$ $$e\left(\frac{89}{105}\right)$$
$$\chi_{1225}(39,\cdot)$$ 1225.bt 210 yes $$1$$ $$1$$ $$e\left(\frac{73}{210}\right)$$ $$e\left(\frac{191}{210}\right)$$ $$e\left(\frac{73}{105}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{3}{70}\right)$$ $$e\left(\frac{86}{105}\right)$$ $$e\left(\frac{19}{105}\right)$$ $$e\left(\frac{127}{210}\right)$$ $$e\left(\frac{29}{70}\right)$$ $$e\left(\frac{41}{105}\right)$$
$$\chi_{1225}(41,\cdot)$$ 1225.bl 70 yes $$-1$$ $$1$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{53}{70}\right)$$ $$e\left(\frac{34}{35}\right)$$ $$e\left(\frac{17}{70}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{51}{70}\right)$$ $$e\left(\frac{41}{70}\right)$$ $$e\left(\frac{33}{35}\right)$$
$$\chi_{1225}(43,\cdot)$$ 1225.y 28 no $$-1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{1225}(44,\cdot)$$ 1225.bt 210 yes $$1$$ $$1$$ $$e\left(\frac{179}{210}\right)$$ $$e\left(\frac{103}{210}\right)$$ $$e\left(\frac{74}{105}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{39}{70}\right)$$ $$e\left(\frac{103}{105}\right)$$ $$e\left(\frac{2}{105}\right)$$ $$e\left(\frac{41}{210}\right)$$ $$e\left(\frac{27}{70}\right)$$ $$e\left(\frac{43}{105}\right)$$
$$\chi_{1225}(46,\cdot)$$ 1225.bo 105 yes $$1$$ $$1$$ $$e\left(\frac{23}{105}\right)$$ $$e\left(\frac{76}{105}\right)$$ $$e\left(\frac{46}{105}\right)$$ $$e\left(\frac{33}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{47}{105}\right)$$ $$e\left(\frac{58}{105}\right)$$ $$e\left(\frac{17}{105}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{92}{105}\right)$$