LMFDB - Dirichlet character orbit 85.r

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, base_ring=CyclotomicField(16))

M = H._module

chi = DirichletCharacter(H, M([4,13]))

chi.galois_orbit()

[g,chi] = znchar(Mod(12,85))

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Basic properties

Modulus:	$85$
Conductor:	$85$	sage: chi.conductor() pari: znconreyconductor(g,chi)
Order:	$16$	sage: chi.multiplicative_order() pari: charorder(g,chi)
Real:	no
Primitive:	yes	sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1
Minimal:	yes
Parity:	even	sage: chi.is_odd() pari: zncharisodd(g,chi)

Related number fields

Field of values:	$\Q(\zeta_{16})$
Fixed field:	16.16.698833752810013621337890625.1

Characters in Galois orbit

Character	$-1$	$1$	$2$	$3$	$4$	$6$	$7$	$8$	$9$	$11$	$12$	$13$
$\chi_{85}(12,\cdot)$	$1$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{9}{16}\right)$	$i$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{13}{16}\right)$	$1$
$\chi_{85}(22,\cdot)$	$1$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{1}{16}\right)$	$i$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{5}{16}\right)$	$1$
$\chi_{85}(23,\cdot)$	$1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{3}{16}\right)$	$-i$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{15}{16}\right)$	$1$
$\chi_{85}(28,\cdot)$	$1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{11}{16}\right)$	$-i$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{7}{16}\right)$	$1$
$\chi_{85}(37,\cdot)$	$1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{13}{16}\right)$	$i$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{1}{16}\right)$	$1$
$\chi_{85}(58,\cdot)$	$1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{15}{16}\right)$	$-i$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{11}{16}\right)$	$1$
$\chi_{85}(78,\cdot)$	$1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{7}{16}\right)$	$-i$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{3}{16}\right)$	$1$
$\chi_{85}(82,\cdot)$	$1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{16}\right)$	$i$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{9}{16}\right)$	$1$

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Characters in Galois orbit

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	85.r
Modulus	$85$
Conductor	$85$
Order	$16$
Real	no
Primitive	yes
Minimal	yes
Parity	even