# Properties

 Modulus $51$ Structure $$C_{2}\times C_{16}$$ Order $32$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(51)

pari: g = idealstar(,51,2)

## Character group

 sage: G.order()  pari: g.no Order = 32 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{16}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{51}(35,\cdot)$, $\chi_{51}(37,\cdot)$

## First 32 of 32 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{51}(1,\cdot)$$ 51.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{51}(2,\cdot)$$ 51.g 8 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$1$$
$$\chi_{51}(4,\cdot)$$ 51.e 4 no $$1$$ $$1$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$1$$ $$-i$$ $$1$$
$$\chi_{51}(5,\cdot)$$ 51.i 16 yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$-1$$
$$\chi_{51}(7,\cdot)$$ 51.j 16 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$
$$\chi_{51}(8,\cdot)$$ 51.g 8 yes $$-1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{51}(10,\cdot)$$ 51.j 16 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$-1$$
$$\chi_{51}(11,\cdot)$$ 51.i 16 yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$-1$$
$$\chi_{51}(13,\cdot)$$ 51.e 4 no $$1$$ $$1$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$1$$ $$i$$ $$1$$
$$\chi_{51}(14,\cdot)$$ 51.i 16 yes $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$-1$$
$$\chi_{51}(16,\cdot)$$ 51.d 2 no $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{51}(19,\cdot)$$ 51.h 8 no $$1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$1$$
$$\chi_{51}(20,\cdot)$$ 51.i 16 yes $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{1}{16}\right)$$ $$-1$$
$$\chi_{51}(22,\cdot)$$ 51.j 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$ $$-1$$
$$\chi_{51}(23,\cdot)$$ 51.i 16 yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
$$\chi_{51}(25,\cdot)$$ 51.h 8 no $$1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$1$$
$$\chi_{51}(26,\cdot)$$ 51.g 8 yes $$-1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$1$$
$$\chi_{51}(28,\cdot)$$ 51.j 16 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
$$\chi_{51}(29,\cdot)$$ 51.i 16 yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$ $$-1$$
$$\chi_{51}(31,\cdot)$$ 51.j 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{1}{16}\right)$$ $$-1$$
$$\chi_{51}(32,\cdot)$$ 51.g 8 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$1$$
$$\chi_{51}(35,\cdot)$$ 51.b 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{51}(37,\cdot)$$ 51.j 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$-1$$
$$\chi_{51}(38,\cdot)$$ 51.f 4 yes $$-1$$ $$1$$ $$1$$ $$1$$ $$i$$ $$i$$ $$1$$ $$i$$ $$-i$$ $$1$$ $$i$$ $$1$$
$$\chi_{51}(40,\cdot)$$ 51.j 16 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$-1$$
$$\chi_{51}(41,\cdot)$$ 51.i 16 yes $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$-1$$
$$\chi_{51}(43,\cdot)$$ 51.h 8 no $$1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{51}(44,\cdot)$$ 51.i 16 yes $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$
$$\chi_{51}(46,\cdot)$$ 51.j 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$-1$$
$$\chi_{51}(47,\cdot)$$ 51.f 4 yes $$-1$$ $$1$$ $$1$$ $$1$$ $$-i$$ $$-i$$ $$1$$ $$-i$$ $$i$$ $$1$$ $$-i$$ $$1$$
$$\chi_{51}(49,\cdot)$$ 51.h 8 no $$1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$1$$
$$\chi_{51}(50,\cdot)$$ 51.c 2 yes $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$