Properties

Label 47.c
Modulus $47$
Conductor $47$
Order $23$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(47, base_ring=CyclotomicField(46))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(2,47))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(47\)
Conductor: \(47\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
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: \(\Q(\zeta_{23})\)
Fixed field: \(\Q(\zeta_{47})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{47}(2,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{17}{23}\right)\)
\(\chi_{47}(3,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{1}{23}\right)\)
\(\chi_{47}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{11}{23}\right)\)
\(\chi_{47}(6,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{18}{23}\right)\)
\(\chi_{47}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{20}{23}\right)\)
\(\chi_{47}(8,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{5}{23}\right)\)
\(\chi_{47}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{2}{23}\right)\)
\(\chi_{47}(12,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{12}{23}\right)\)
\(\chi_{47}(14,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{14}{23}\right)\)
\(\chi_{47}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{22}{23}\right)\)
\(\chi_{47}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{10}{23}\right)\)
\(\chi_{47}(18,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{19}{23}\right)\)
\(\chi_{47}(21,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{21}{23}\right)\)
\(\chi_{47}(24,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{6}{23}\right)\)
\(\chi_{47}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{7}{23}\right)\)
\(\chi_{47}(27,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{3}{23}\right)\)
\(\chi_{47}(28,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{8}{23}\right)\)
\(\chi_{47}(32,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{16}{23}\right)\)
\(\chi_{47}(34,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{4}{23}\right)\)
\(\chi_{47}(36,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{13}{23}\right)\)
\(\chi_{47}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{9}{23}\right)\)
\(\chi_{47}(42,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{15}{23}\right)\)