Properties

Label 675.bf
Modulus $675$
Conductor $675$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(675, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([65,72]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(11,675))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(675\)
Conductor: \(675\)
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 \(-1\) \(1\) \(2\) \(4\) \(7\) \(8\) \(11\) \(13\) \(14\) \(16\) \(17\) \(19\)
\(\chi_{675}(11,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{675}(41,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{675}(56,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{675}(86,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{675}(131,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{675}(146,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{675}(191,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{675}(221,\cdot)\) \(-1\) \(1\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{675}(236,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{675}(266,\cdot)\) \(-1\) \(1\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{675}(281,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{675}(311,\cdot)\) \(-1\) \(1\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{675}(356,\cdot)\) \(-1\) \(1\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{675}(371,\cdot)\) \(-1\) \(1\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{675}(416,\cdot)\) \(-1\) \(1\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{675}(446,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{675}(461,\cdot)\) \(-1\) \(1\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{675}(491,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{675}(506,\cdot)\) \(-1\) \(1\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{675}(536,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{675}(581,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{675}(596,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{675}(641,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{675}(671,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{2}{15}\right)\)