Properties

Modulus 36
Structure \(C_{6}\times C_{2}\)
Order 12

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(36)
pari: g = idealstar(,36,2)

Character group

sage: G.order()
pari: g.no
Order = 12
sage: H.invariants()
pari: g.cyc
Structure = \(C_{6}\times C_{2}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{36}(29,\cdot)$, $\chi_{36}(19,\cdot)$

Characters

order primitive 1 5 7 11 13 17 19 23 25 29 31 35
\(\chi_{36}(1,\cdot)\) 1 No 1 1 1 1 1 1 1 1 1 1 1 1
\(\chi_{36}(5,\cdot)\) 6 No 1 \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) 1 \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\)
\(\chi_{36}(7,\cdot)\) 6 Yes 1 \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) 1 \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\)
\(\chi_{36}(11,\cdot)\) 6 Yes 1 \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) 1
\(\chi_{36}(13,\cdot)\) 3 No 1 \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) 1 1 \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) 1
\(\chi_{36}(17,\cdot)\) 2 No 1 \(-1\) 1 \(-1\) 1 \(-1\) 1 \(-1\) 1 \(-1\) 1 \(-1\)
\(\chi_{36}(19,\cdot)\) 2 No 1 1 \(-1\) \(-1\) 1 1 \(-1\) \(-1\) 1 1 \(-1\) \(-1\)
\(\chi_{36}(23,\cdot)\) 6 Yes 1 \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) 1
\(\chi_{36}(25,\cdot)\) 3 No 1 \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) 1 1 \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) 1
\(\chi_{36}(29,\cdot)\) 6 No 1 \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) 1 \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\)
\(\chi_{36}(31,\cdot)\) 6 Yes 1 \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) 1 \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\)
\(\chi_{36}(35,\cdot)\) 2 No 1 \(-1\) \(-1\) 1 1 \(-1\) \(-1\) 1 1 \(-1\) \(-1\) 1