# Properties

 Modulus $70$ Structure $$C_{2}\times C_{12}$$ Order $24$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(70)

pari: g = idealstar(,70,2)

## Character group

 sage: G.order()  pari: g.no Order = 24 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{12}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{70}(57,\cdot)$, $\chi_{70}(31,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$27$$ $$29$$ $$31$$
$$\chi_{70}(1,\cdot)$$ 70.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{70}(3,\cdot)$$ 70.k 12 no $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{70}(9,\cdot)$$ 70.i 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{70}(11,\cdot)$$ 70.e 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{70}(13,\cdot)$$ 70.g 4 no $$1$$ $$1$$ $$-i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-1$$
$$\chi_{70}(17,\cdot)$$ 70.k 12 no $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{70}(19,\cdot)$$ 70.h 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{70}(23,\cdot)$$ 70.l 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{70}(27,\cdot)$$ 70.g 4 no $$1$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$-1$$
$$\chi_{70}(29,\cdot)$$ 70.c 2 no $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$
$$\chi_{70}(31,\cdot)$$ 70.j 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{70}(33,\cdot)$$ 70.k 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{70}(37,\cdot)$$ 70.l 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{70}(39,\cdot)$$ 70.i 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{70}(41,\cdot)$$ 70.b 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
$$\chi_{70}(43,\cdot)$$ 70.f 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$i$$ $$-i$$ $$-1$$ $$1$$
$$\chi_{70}(47,\cdot)$$ 70.k 12 no $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{70}(51,\cdot)$$ 70.e 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{70}(53,\cdot)$$ 70.l 12 no $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{70}(57,\cdot)$$ 70.f 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-i$$ $$i$$ $$-1$$ $$1$$
$$\chi_{70}(59,\cdot)$$ 70.h 6 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{70}(61,\cdot)$$ 70.j 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{70}(67,\cdot)$$ 70.l 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{70}(69,\cdot)$$ 70.d 2 no $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$