# Properties

 Modulus $78$ Structure $$C_{2}\times C_{12}$$ Order $24$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(78)

pari: g = idealstar(,78,2)

## 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 = $\chi_{78}(53,\cdot)$, $\chi_{78}(67,\cdot)$

## 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$
$$\chi_{78}(1,\cdot)$$ 78.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{78}(5,\cdot)$$ 78.g 4 no $$1$$ $$1$$ $$i$$ $$i$$ $$-i$$ $$1$$ $$-i$$ $$1$$ $$-1$$ $$-1$$ $$-i$$ $$-1$$
$$\chi_{78}(7,\cdot)$$ 78.l 12 no $$-1$$ $$1$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{78}(11,\cdot)$$ 78.k 12 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{78}(17,\cdot)$$ 78.j 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{78}(19,\cdot)$$ 78.l 12 no $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{78}(23,\cdot)$$ 78.j 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{78}(25,\cdot)$$ 78.b 2 no $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{78}(29,\cdot)$$ 78.h 6 no $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{78}(31,\cdot)$$ 78.f 4 no $$-1$$ $$1$$ $$-i$$ $$i$$ $$i$$ $$-1$$ $$-i$$ $$-1$$ $$-1$$ $$1$$ $$-i$$ $$1$$
$$\chi_{78}(35,\cdot)$$ 78.h 6 no $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{78}(37,\cdot)$$ 78.l 12 no $$-1$$ $$1$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{78}(41,\cdot)$$ 78.k 12 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{78}(43,\cdot)$$ 78.i 6 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{78}(47,\cdot)$$ 78.g 4 no $$1$$ $$1$$ $$-i$$ $$-i$$ $$i$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$-1$$ $$i$$ $$-1$$
$$\chi_{78}(49,\cdot)$$ 78.i 6 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{78}(53,\cdot)$$ 78.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
$$\chi_{78}(55,\cdot)$$ 78.e 3 no $$1$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{78}(59,\cdot)$$ 78.k 12 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{78}(61,\cdot)$$ 78.e 3 no $$1$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{78}(67,\cdot)$$ 78.l 12 no $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{78}(71,\cdot)$$ 78.k 12 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{78}(73,\cdot)$$ 78.f 4 no $$-1$$ $$1$$ $$i$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$-1$$ $$-1$$ $$1$$ $$i$$ $$1$$
$$\chi_{78}(77,\cdot)$$ 78.d 2 no $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$