# Properties

 Modulus $230$ Structure $$C_{2}\times C_{44}$$ Order $88$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(230)

pari: g = idealstar(,230,2)

## Character group

 sage: G.order()  pari: g.no Order = 88 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{44}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{230}(47,\cdot)$, $\chi_{230}(51,\cdot)$

## First 32 of 88 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 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$27$$ $$29$$
$$\chi_{230}(1,\cdot)$$ 230.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{230}(3,\cdot)$$ 230.k 44 no $$-1$$ $$1$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{230}(7,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{230}(9,\cdot)$$ 230.j 22 no $$1$$ $$1$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$
$$\chi_{230}(11,\cdot)$$ 230.h 22 no $$-1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{230}(13,\cdot)$$ 230.k 44 no $$-1$$ $$1$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{230}(17,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{230}(19,\cdot)$$ 230.i 22 no $$-1$$ $$1$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{230}(21,\cdot)$$ 230.h 22 no $$-1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$
$$\chi_{230}(27,\cdot)$$ 230.k 44 no $$-1$$ $$1$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{230}(29,\cdot)$$ 230.j 22 no $$1$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{230}(31,\cdot)$$ 230.g 11 no $$1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{230}(33,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{230}(37,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{230}(39,\cdot)$$ 230.j 22 no $$1$$ $$1$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{230}(41,\cdot)$$ 230.g 11 no $$1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{230}(43,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{230}(47,\cdot)$$ 230.f 4 no $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$i$$ $$-1$$
$$\chi_{230}(49,\cdot)$$ 230.j 22 no $$1$$ $$1$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{230}(51,\cdot)$$ 230.h 22 no $$-1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{230}(53,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{230}(57,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{230}(59,\cdot)$$ 230.j 22 no $$1$$ $$1$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{230}(61,\cdot)$$ 230.h 22 no $$-1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{230}(63,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{230}(67,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{230}(71,\cdot)$$ 230.g 11 no $$1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$
$$\chi_{230}(73,\cdot)$$ 230.k 44 no $$-1$$ $$1$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{230}(77,\cdot)$$ 230.k 44 no $$-1$$ $$1$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{230}(79,\cdot)$$ 230.i 22 no $$-1$$ $$1$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{230}(81,\cdot)$$ 230.g 11 no $$1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{230}(83,\cdot)$$ 230.l 44 no $$1$$ $$1$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$