Properties

Label 475.bf
Modulus $475$
Conductor $475$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(475, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([54,5]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(21,475))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(475\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{475}(21,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{61}{90}\right)\)
\(\chi_{475}(41,\cdot)\) \(-1\) \(1\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{37}{90}\right)\)
\(\chi_{475}(71,\cdot)\) \(-1\) \(1\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{31}{90}\right)\)
\(\chi_{475}(86,\cdot)\) \(-1\) \(1\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{83}{90}\right)\)
\(\chi_{475}(91,\cdot)\) \(-1\) \(1\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{77}{90}\right)\)
\(\chi_{475}(116,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{90}\right)\)
\(\chi_{475}(136,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{73}{90}\right)\)
\(\chi_{475}(146,\cdot)\) \(-1\) \(1\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{71}{90}\right)\)
\(\chi_{475}(166,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{67}{90}\right)\)
\(\chi_{475}(181,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{29}{90}\right)\)
\(\chi_{475}(186,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{23}{90}\right)\)
\(\chi_{475}(211,\cdot)\) \(-1\) \(1\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{43}{90}\right)\)
\(\chi_{475}(231,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{19}{90}\right)\)
\(\chi_{475}(241,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{17}{90}\right)\)
\(\chi_{475}(261,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{13}{90}\right)\)
\(\chi_{475}(281,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{59}{90}\right)\)
\(\chi_{475}(306,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{79}{90}\right)\)
\(\chi_{475}(336,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{53}{90}\right)\)
\(\chi_{475}(356,\cdot)\) \(-1\) \(1\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{49}{90}\right)\)
\(\chi_{475}(371,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{11}{90}\right)\)
\(\chi_{475}(421,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{90}\right)\)
\(\chi_{475}(431,\cdot)\) \(-1\) \(1\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{89}{90}\right)\)
\(\chi_{475}(466,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{47}{90}\right)\)
\(\chi_{475}(471,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{41}{90}\right)\)