Properties

 Modulus $68$ Structure $$C_{2}\times C_{16}$$ Order $32$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(68)

pari: g = idealstar(,68,2)

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 = $\chi_{68}(35,\cdot)$, $\chi_{68}(37,\cdot)$

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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$19$$ $$21$$ $$23$$
$$\chi_{68}(1,\cdot)$$ 68.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{68}(3,\cdot)$$ 68.i 16 yes $$1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$
$$\chi_{68}(5,\cdot)$$ 68.j 16 no $$-1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$
$$\chi_{68}(7,\cdot)$$ 68.i 16 yes $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{68}(9,\cdot)$$ 68.h 8 no $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{68}(11,\cdot)$$ 68.i 16 yes $$1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{68}(13,\cdot)$$ 68.e 4 no $$1$$ $$1$$ $$i$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-i$$
$$\chi_{68}(15,\cdot)$$ 68.g 8 yes $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{68}(19,\cdot)$$ 68.g 8 yes $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{68}(21,\cdot)$$ 68.e 4 no $$1$$ $$1$$ $$-i$$ $$-i$$ $$i$$ $$-1$$ $$i$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$i$$
$$\chi_{68}(23,\cdot)$$ 68.i 16 yes $$1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$
$$\chi_{68}(25,\cdot)$$ 68.h 8 no $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{68}(27,\cdot)$$ 68.i 16 yes $$1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$
$$\chi_{68}(29,\cdot)$$ 68.j 16 no $$-1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{68}(31,\cdot)$$ 68.i 16 yes $$1$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$
$$\chi_{68}(33,\cdot)$$ 68.b 2 no $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$
$$\chi_{68}(35,\cdot)$$ 68.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$
$$\chi_{68}(37,\cdot)$$ 68.j 16 no $$-1$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$
$$\chi_{68}(39,\cdot)$$ 68.i 16 yes $$1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{68}(41,\cdot)$$ 68.j 16 no $$-1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$
$$\chi_{68}(43,\cdot)$$ 68.g 8 yes $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{68}(45,\cdot)$$ 68.j 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$
$$\chi_{68}(47,\cdot)$$ 68.f 4 yes $$-1$$ $$1$$ $$-i$$ $$i$$ $$i$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$1$$ $$1$$ $$i$$
$$\chi_{68}(49,\cdot)$$ 68.h 8 no $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{68}(53,\cdot)$$ 68.h 8 no $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{68}(55,\cdot)$$ 68.f 4 yes $$-1$$ $$1$$ $$i$$ $$-i$$ $$-i$$ $$-1$$ $$-i$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-i$$
$$\chi_{68}(57,\cdot)$$ 68.j 16 no $$-1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{68}(59,\cdot)$$ 68.g 8 yes $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{68}(61,\cdot)$$ 68.j 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{68}(63,\cdot)$$ 68.i 16 yes $$1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$
$$\chi_{68}(65,\cdot)$$ 68.j 16 no $$-1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$
$$\chi_{68}(67,\cdot)$$ 68.d 2 yes $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$