# Properties

 Modulus $205$ Structure $$C_{40}\times C_{4}$$ Order $160$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(205)

pari: g = idealstar(,205,2)

## Character group

 sage: G.order()  pari: g.no Order = 160 sage: H.invariants()  pari: g.cyc Structure = $$C_{40}\times C_{4}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{205}(42,\cdot)$, $\chi_{205}(6,\cdot)$

## First 32 of 160 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$
$$\chi_{205}(1,\cdot)$$ 205.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{205}(2,\cdot)$$ 205.t 20 yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{205}(3,\cdot)$$ 205.l 8 yes $$1$$ $$1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{205}(4,\cdot)$$ 205.r 10 yes $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{205}(6,\cdot)$$ 205.z 40 no $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$
$$\chi_{205}(7,\cdot)$$ 205.y 40 yes $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-i$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$
$$\chi_{205}(8,\cdot)$$ 205.t 20 yes $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{205}(9,\cdot)$$ 205.j 4 yes $$1$$ $$1$$ $$1$$ $$-i$$ $$1$$ $$-i$$ $$-i$$ $$1$$ $$-1$$ $$i$$ $$-i$$ $$-i$$
$$\chi_{205}(11,\cdot)$$ 205.z 40 no $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$i$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$
$$\chi_{205}(12,\cdot)$$ 205.y 40 yes $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-i$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$
$$\chi_{205}(13,\cdot)$$ 205.bb 40 yes $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-i$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$
$$\chi_{205}(14,\cdot)$$ 205.m 8 yes $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{205}(16,\cdot)$$ 205.k 5 no $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{205}(17,\cdot)$$ 205.bb 40 yes $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$i$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$
$$\chi_{205}(18,\cdot)$$ 205.u 20 yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$
$$\chi_{205}(19,\cdot)$$ 205.ba 40 yes $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$-i$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$
$$\chi_{205}(21,\cdot)$$ 205.x 20 no $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$
$$\chi_{205}(22,\cdot)$$ 205.bb 40 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$i$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$
$$\chi_{205}(23,\cdot)$$ 205.v 20 yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$-1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$
$$\chi_{205}(24,\cdot)$$ 205.ba 40 yes $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$-i$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$
$$\chi_{205}(26,\cdot)$$ 205.z 40 no $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$
$$\chi_{205}(27,\cdot)$$ 205.l 8 yes $$1$$ $$1$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{205}(28,\cdot)$$ 205.bb 40 yes $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-i$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$
$$\chi_{205}(29,\cdot)$$ 205.ba 40 yes $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$i$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$
$$\chi_{205}(31,\cdot)$$ 205.p 10 no $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{205}(32,\cdot)$$ 205.f 4 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-1$$ $$i$$ $$1$$ $$i$$ $$1$$ $$-i$$ $$1$$ $$-1$$
$$\chi_{205}(33,\cdot)$$ 205.w 20 yes $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{205}(34,\cdot)$$ 205.ba 40 yes $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$i$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$
$$\chi_{205}(36,\cdot)$$ 205.x 20 no $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$-i$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$
$$\chi_{205}(37,\cdot)$$ 205.u 20 yes $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$-i$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$
$$\chi_{205}(38,\cdot)$$ 205.l 8 yes $$1$$ $$1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{205}(39,\cdot)$$ 205.s 20 yes $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$-i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$