Properties

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

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Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(68)
pari: g = idealstar(,68,2)

Character group

sage: G.order()
pari: g.no
Order = 32
sage: H.invariants()
pari: g.cyc
Structure = \(C_{16}\times C_{2}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{68}(37,\cdot)$, $\chi_{68}(35,\cdot)$

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.

orbit label 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 \(-1\) \(1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\)