# Properties

 Modulus $37$ Structure $$C_{36}$$ Order $36$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(37)

pari: g = idealstar(,37,2)

## Character group

 sage: G.order()  pari: g.no Order = 36 sage: H.invariants()  pari: g.cyc Structure = $$C_{36}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{37}(2,\cdot)$

## First 32 of 36 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{37}(1,\cdot)$$ 37.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{37}(2,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$-i$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{37}(3,\cdot)$$ 37.h 18 yes $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$-1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{37}(4,\cdot)$$ 37.h 18 yes $$1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$-1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{37}(5,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$i$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{37}(6,\cdot)$$ 37.d 4 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-1$$ $$i$$ $$i$$ $$1$$ $$i$$ $$1$$ $$1$$ $$-1$$
$$\chi_{37}(7,\cdot)$$ 37.f 9 yes $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{37}(8,\cdot)$$ 37.g 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$-1$$
$$\chi_{37}(9,\cdot)$$ 37.f 9 yes $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{37}(10,\cdot)$$ 37.c 3 yes $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$
$$\chi_{37}(11,\cdot)$$ 37.e 6 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$
$$\chi_{37}(12,\cdot)$$ 37.f 9 yes $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{37}(13,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$i$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{37}(14,\cdot)$$ 37.g 12 yes $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$
$$\chi_{37}(15,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$-i$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{37}(16,\cdot)$$ 37.f 9 yes $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{37}(17,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$i$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{37}(18,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$-i$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{37}(19,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$i$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{37}(20,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$-i$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{37}(21,\cdot)$$ 37.h 18 yes $$1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$-1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{37}(22,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$i$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{37}(23,\cdot)$$ 37.g 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$
$$\chi_{37}(24,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$-i$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{37}(25,\cdot)$$ 37.h 18 yes $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$-1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{37}(26,\cdot)$$ 37.c 3 yes $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$
$$\chi_{37}(27,\cdot)$$ 37.e 6 yes $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$
$$\chi_{37}(28,\cdot)$$ 37.h 18 yes $$1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$-1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{37}(29,\cdot)$$ 37.g 12 yes $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$-1$$
$$\chi_{37}(30,\cdot)$$ 37.h 18 yes $$1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$-1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{37}(31,\cdot)$$ 37.d 4 yes $$-1$$ $$1$$ $$i$$ $$-1$$ $$-1$$ $$-i$$ $$-i$$ $$1$$ $$-i$$ $$1$$ $$1$$ $$-1$$
$$\chi_{37}(32,\cdot)$$ 37.i 36 yes $$-1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$-i$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$