Properties

Modulus 35
Structure \(C_{12}\times C_{2}\)
Order 24

Learn more about

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(35)
pari: g = idealstar(,35,2)

Character group

sage: G.order()
pari: g.no
Order = 24
sage: H.invariants()
pari: g.cyc
Structure = \(C_{12}\times C_{2}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{35}(3,\cdot)$, $\chi_{35}(34,\cdot)$

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 8 9 11 12 13 16
\(\chi_{35}(1,\cdot)\) 35.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{35}(2,\cdot)\) 35.l 12 Yes \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{35}(3,\cdot)\) 35.k 12 Yes \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{35}(4,\cdot)\) 35.j 6 Yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{35}(6,\cdot)\) 35.d 2 No \(-1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(-1\) \(1\)
\(\chi_{35}(8,\cdot)\) 35.g 4 No \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(1\) \(i\) \(-1\) \(1\) \(-i\) \(i\) \(1\)
\(\chi_{35}(9,\cdot)\) 35.j 6 Yes \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{35}(11,\cdot)\) 35.e 3 No \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{35}(12,\cdot)\) 35.k 12 Yes \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{35}(13,\cdot)\) 35.f 4 Yes \(1\) \(1\) \(-i\) \(-i\) \(-1\) \(-1\) \(i\) \(-1\) \(1\) \(i\) \(-i\) \(1\)
\(\chi_{35}(16,\cdot)\) 35.e 3 No \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{35}(17,\cdot)\) 35.k 12 Yes \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{35}(18,\cdot)\) 35.l 12 Yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{35}(19,\cdot)\) 35.i 6 Yes \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{35}(22,\cdot)\) 35.g 4 No \(-1\) \(1\) \(i\) \(-i\) \(-1\) \(1\) \(-i\) \(-1\) \(1\) \(i\) \(-i\) \(1\)
\(\chi_{35}(23,\cdot)\) 35.l 12 Yes \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{35}(24,\cdot)\) 35.i 6 Yes \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{35}(26,\cdot)\) 35.h 6 No \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{35}(27,\cdot)\) 35.f 4 Yes \(1\) \(1\) \(i\) \(i\) \(-1\) \(-1\) \(-i\) \(-1\) \(1\) \(-i\) \(i\) \(1\)
\(\chi_{35}(29,\cdot)\) 35.b 2 No \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\)
\(\chi_{35}(31,\cdot)\) 35.h 6 No \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{35}(32,\cdot)\) 35.l 12 Yes \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{35}(33,\cdot)\) 35.k 12 Yes \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{35}(34,\cdot)\) 35.c 2 Yes \(-1\) \(1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(1\) \(1\) \(1\)