# Properties

 Modulus $32$ Structure $$C_{2}\times C_{8}$$ Order $16$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(32)

pari: g = idealstar(,32,2)

## Character group

 sage: G.order()  pari: g.no Order = 16 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{8}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{32}(31,\cdot)$, $\chi_{32}(5,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{32}(1,\cdot)$$ 32.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{32}(3,\cdot)$$ 32.h 8 yes $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{32}(5,\cdot)$$ 32.g 8 yes $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{32}(7,\cdot)$$ 32.f 4 no $$-1$$ $$1$$ $$i$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$i$$
$$\chi_{32}(9,\cdot)$$ 32.e 4 no $$1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$i$$ $$1$$ $$1$$ $$i$$ $$-i$$
$$\chi_{32}(11,\cdot)$$ 32.h 8 yes $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{32}(13,\cdot)$$ 32.g 8 yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{32}(15,\cdot)$$ 32.d 2 no $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$
$$\chi_{32}(17,\cdot)$$ 32.b 2 no $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
$$\chi_{32}(19,\cdot)$$ 32.h 8 yes $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{32}(21,\cdot)$$ 32.g 8 yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{32}(23,\cdot)$$ 32.f 4 no $$-1$$ $$1$$ $$-i$$ $$-i$$ $$1$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$-i$$
$$\chi_{32}(25,\cdot)$$ 32.e 4 no $$1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$-i$$ $$1$$ $$1$$ $$-i$$ $$i$$
$$\chi_{32}(27,\cdot)$$ 32.h 8 yes $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{32}(29,\cdot)$$ 32.g 8 yes $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{32}(31,\cdot)$$ 32.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$