Properties

Modulus 85
Structure \(C_{16}\times C_{4}\)
Order 64

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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(85)
pari: g = idealstar(,85,2)

Character group

sage: G.order()
pari: g.no
Order = 64
sage: H.invariants()
pari: g.cyc
Structure = \(C_{16}\times C_{4}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{85}(71,\cdot)$, $\chi_{85}(52,\cdot)$

First 32 of 64 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 2 3 4 6 7 8 9 11 12 13
\(\chi_{85}(1,\cdot)\) 85.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{85}(2,\cdot)\) 85.k 8 Yes \(-1\) \(1\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(i\)
\(\chi_{85}(3,\cdot)\) 85.o 16 Yes \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(-1\)
\(\chi_{85}(4,\cdot)\) 85.j 4 Yes \(1\) \(1\) \(1\) \(i\) \(1\) \(i\) \(-i\) \(1\) \(-1\) \(i\) \(i\) \(-1\)
\(\chi_{85}(6,\cdot)\) 85.q 16 No \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(-i\)
\(\chi_{85}(7,\cdot)\) 85.o 16 Yes \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(-i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(-1\)
\(\chi_{85}(8,\cdot)\) 85.k 8 Yes \(-1\) \(1\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(-i\)
\(\chi_{85}(9,\cdot)\) 85.m 8 Yes \(1\) \(1\) \(i\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(1\)
\(\chi_{85}(11,\cdot)\) 85.q 16 No \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(-i\)
\(\chi_{85}(12,\cdot)\) 85.r 16 Yes \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(1\)
\(\chi_{85}(13,\cdot)\) 85.f 4 Yes \(-1\) \(1\) \(i\) \(-1\) \(-1\) \(-i\) \(-1\) \(-i\) \(1\) \(-i\) \(1\) \(i\)
\(\chi_{85}(14,\cdot)\) 85.p 16 Yes \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(-i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(-i\)
\(\chi_{85}(16,\cdot)\) 85.d 2 No \(1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\)
\(\chi_{85}(18,\cdot)\) 85.h 4 No \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(1\) \(-i\) \(i\)
\(\chi_{85}(19,\cdot)\) 85.m 8 Yes \(1\) \(1\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(i\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(1\)
\(\chi_{85}(21,\cdot)\) 85.e 4 No \(1\) \(1\) \(-1\) \(-i\) \(1\) \(i\) \(i\) \(-1\) \(-1\) \(i\) \(-i\) \(1\)
\(\chi_{85}(22,\cdot)\) 85.r 16 Yes \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(1\)
\(\chi_{85}(23,\cdot)\) 85.r 16 Yes \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(-i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(1\)
\(\chi_{85}(24,\cdot)\) 85.p 16 Yes \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\)
\(\chi_{85}(26,\cdot)\) 85.l 8 No \(1\) \(1\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\) \(i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-1\)
\(\chi_{85}(27,\cdot)\) 85.o 16 Yes \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(-i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(-1\)
\(\chi_{85}(28,\cdot)\) 85.r 16 Yes \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(-i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(1\)
\(\chi_{85}(29,\cdot)\) 85.p 16 Yes \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(-i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(-i\)
\(\chi_{85}(31,\cdot)\) 85.q 16 No \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(-i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(i\)
\(\chi_{85}(32,\cdot)\) 85.k 8 Yes \(-1\) \(1\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(i\)
\(\chi_{85}(33,\cdot)\) 85.g 4 Yes \(-1\) \(1\) \(-i\) \(-i\) \(-1\) \(-1\) \(i\) \(i\) \(-1\) \(-1\) \(i\) \(i\)
\(\chi_{85}(36,\cdot)\) 85.l 8 No \(1\) \(1\) \(i\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(-1\)
\(\chi_{85}(37,\cdot)\) 85.r 16 Yes \(1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(1\)
\(\chi_{85}(38,\cdot)\) 85.i 4 Yes \(-1\) \(1\) \(i\) \(1\) \(-1\) \(i\) \(1\) \(-i\) \(1\) \(i\) \(-1\) \(i\)
\(\chi_{85}(39,\cdot)\) 85.p 16 Yes \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(-i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(-i\)
\(\chi_{85}(41,\cdot)\) 85.q 16 No \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(-i\)
\(\chi_{85}(42,\cdot)\) 85.n 8 Yes \(-1\) \(1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\)