Properties

Label 231.bf
Modulus $231$
Conductor $231$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,5,27]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(17,231))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(231\)
Conductor: \(231\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(13\) \(16\) \(17\) \(19\) \(20\)
\(\chi_{231}(17,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{231}(68,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{231}(101,\cdot)\) \(-1\) \(1\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{231}(173,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{231}(194,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{231}(206,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{231}(215,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{231}(227,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{5}\right)\)