Properties

Modulus 65
Structure \(C_{12}\times C_{4}\)
Order 48

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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(65)
pari: g = idealstar(,65,2)

Character group

sage: G.order()
pari: g.no
Order = 48
sage: H.invariants()
pari: g.cyc
Structure = \(C_{12}\times C_{4}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{65}(41,\cdot)$, $\chi_{65}(27,\cdot)$

First 32 of 48 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 14
\(\chi_{65}(1,\cdot)\) 65.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{65}(2,\cdot)\) 65.o 12 Yes \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(-1\)
\(\chi_{65}(3,\cdot)\) 65.q 12 Yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-i\) \(-1\)
\(\chi_{65}(4,\cdot)\) 65.l 6 Yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)
\(\chi_{65}(6,\cdot)\) 65.p 12 No \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(-1\) \(1\)
\(\chi_{65}(7,\cdot)\) 65.t 12 Yes \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(-1\)
\(\chi_{65}(8,\cdot)\) 65.k 4 Yes \(1\) \(1\) \(1\) \(i\) \(1\) \(i\) \(-1\) \(1\) \(-1\) \(-i\) \(i\) \(-1\)
\(\chi_{65}(9,\cdot)\) 65.n 6 Yes \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)
\(\chi_{65}(11,\cdot)\) 65.p 12 No \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(-1\) \(1\)
\(\chi_{65}(12,\cdot)\) 65.h 4 Yes \(-1\) \(1\) \(-i\) \(-i\) \(-1\) \(-1\) \(-i\) \(i\) \(-1\) \(-1\) \(i\) \(-1\)
\(\chi_{65}(14,\cdot)\) 65.b 2 No \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(1\)
\(\chi_{65}(16,\cdot)\) 65.e 3 No \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)
\(\chi_{65}(17,\cdot)\) 65.r 12 Yes \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(-1\)
\(\chi_{65}(18,\cdot)\) 65.f 4 Yes \(1\) \(1\) \(-1\) \(i\) \(1\) \(-i\) \(1\) \(-1\) \(-1\) \(i\) \(i\) \(-1\)
\(\chi_{65}(19,\cdot)\) 65.s 12 Yes \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(1\) \(1\)
\(\chi_{65}(21,\cdot)\) 65.j 4 No \(-1\) \(1\) \(i\) \(1\) \(-1\) \(i\) \(-i\) \(-i\) \(1\) \(-i\) \(-1\) \(1\)
\(\chi_{65}(22,\cdot)\) 65.q 12 Yes \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(i\) \(-1\)
\(\chi_{65}(23,\cdot)\) 65.r 12 Yes \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(-1\)
\(\chi_{65}(24,\cdot)\) 65.s 12 Yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(1\) \(1\)
\(\chi_{65}(27,\cdot)\) 65.i 4 No \(-1\) \(1\) \(i\) \(-i\) \(-1\) \(1\) \(i\) \(-i\) \(-1\) \(1\) \(i\) \(-1\)
\(\chi_{65}(28,\cdot)\) 65.t 12 Yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(i\) \(-1\)
\(\chi_{65}(29,\cdot)\) 65.n 6 Yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\)
\(\chi_{65}(31,\cdot)\) 65.j 4 No \(-1\) \(1\) \(-i\) \(1\) \(-1\) \(-i\) \(i\) \(i\) \(1\) \(i\) \(-1\) \(1\)
\(\chi_{65}(32,\cdot)\) 65.o 12 Yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(-1\)
\(\chi_{65}(33,\cdot)\) 65.o 12 Yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(i\) \(-1\)
\(\chi_{65}(34,\cdot)\) 65.g 4 Yes \(-1\) \(1\) \(-i\) \(-1\) \(-1\) \(i\) \(i\) \(i\) \(1\) \(-i\) \(1\) \(1\)
\(\chi_{65}(36,\cdot)\) 65.m 6 No \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)
\(\chi_{65}(37,\cdot)\) 65.t 12 Yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(-1\)
\(\chi_{65}(38,\cdot)\) 65.h 4 Yes \(-1\) \(1\) \(i\) \(i\) \(-1\) \(-1\) \(i\) \(-i\) \(-1\) \(-1\) \(-i\) \(-1\)
\(\chi_{65}(41,\cdot)\) 65.p 12 No \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(-1\) \(1\)
\(\chi_{65}(42,\cdot)\) 65.q 12 Yes \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(i\) \(-1\)
\(\chi_{65}(43,\cdot)\) 65.r 12 Yes \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(-1\)