# Properties

 Modulus $273$ Structure $$C_{12}\times C_{6}\times C_{2}$$ Order $144$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(273)

pari: g = idealstar(,273,2)

## Character group

 sage: G.order()  pari: g.no Order = 144 sage: H.invariants()  pari: g.cyc Structure = $$C_{12}\times C_{6}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{273}(92,\cdot)$, $\chi_{273}(157,\cdot)$, $\chi_{273}(106,\cdot)$

## First 32 of 144 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$16$$ $$17$$ $$19$$ $$20$$
$$\chi_{273}(1,\cdot)$$ 273.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{273}(2,\cdot)$$ 273.bv 12 yes $$1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{273}(4,\cdot)$$ 273.t 6 no $$1$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{273}(5,\cdot)$$ 273.cb 12 yes $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$
$$\chi_{273}(8,\cdot)$$ 273.n 4 no $$1$$ $$1$$ $$-i$$ $$-1$$ $$-i$$ $$i$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$i$$ $$i$$
$$\chi_{273}(10,\cdot)$$ 273.bg 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$-1$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{273}(11,\cdot)$$ 273.bw 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$-1$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{273}(16,\cdot)$$ 273.l 3 no $$1$$ $$1$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{273}(17,\cdot)$$ 273.br 6 yes $$1$$ $$1$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{273}(19,\cdot)$$ 273.cg 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$1$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{273}(20,\cdot)$$ 273.ca 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{273}(22,\cdot)$$ 273.k 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{273}(23,\cdot)$$ 273.bp 6 yes $$-1$$ $$1$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{273}(25,\cdot)$$ 273.bj 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$
$$\chi_{273}(29,\cdot)$$ 273.be 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{273}(31,\cdot)$$ 273.bz 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$
$$\chi_{273}(32,\cdot)$$ 273.bv 12 yes $$1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{273}(34,\cdot)$$ 273.p 4 no $$1$$ $$1$$ $$i$$ $$-1$$ $$-i$$ $$-i$$ $$1$$ $$-i$$ $$1$$ $$1$$ $$-i$$ $$i$$
$$\chi_{273}(37,\cdot)$$ 273.bu 12 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{273}(38,\cdot)$$ 273.ba 6 yes $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$
$$\chi_{273}(40,\cdot)$$ 273.bb 6 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$
$$\chi_{273}(41,\cdot)$$ 273.ca 12 yes $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{273}(43,\cdot)$$ 273.bd 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{273}(44,\cdot)$$ 273.cd 12 yes $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$
$$\chi_{273}(46,\cdot)$$ 273.bu 12 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{273}(47,\cdot)$$ 273.cb 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$
$$\chi_{273}(50,\cdot)$$ 273.cc 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{273}(53,\cdot)$$ 273.bk 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$
$$\chi_{273}(55,\cdot)$$ 273.v 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{273}(58,\cdot)$$ 273.bx 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$i$$ $$-1$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{273}(59,\cdot)$$ 273.bs 12 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{273}(61,\cdot)$$ 273.z 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$