Properties

Label 675.bc
Modulus $675$
Conductor $675$
Order $45$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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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([20,18]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(16,675))
 
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: \(45\)
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: $\Q(\zeta_{45})$
Fixed field: 45.45.6444338306249279600681470320699578378786756892621858688031629726344906572421677992679178714752197265625.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(7\) \(8\) \(11\) \(13\) \(14\) \(16\) \(17\) \(19\)
\(\chi_{675}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{675}(31,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{675}(61,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{675}(106,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{675}(121,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{675}(166,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{675}(196,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{675}(211,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{675}(241,\cdot)\) \(1\) \(1\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{675}(256,\cdot)\) \(1\) \(1\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{675}(286,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{675}(331,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{675}(346,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{675}(391,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{675}(421,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{675}(436,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{11}{15}\right)\)
\(\chi_{675}(466,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{4}{15}\right)\)
\(\chi_{675}(481,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{675}(511,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{1}{15}\right)\)
\(\chi_{675}(556,\cdot)\) \(1\) \(1\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{13}{15}\right)\)
\(\chi_{675}(571,\cdot)\) \(1\) \(1\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{2}{15}\right)\)
\(\chi_{675}(616,\cdot)\) \(1\) \(1\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{14}{15}\right)\)
\(\chi_{675}(646,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{7}{15}\right)\)
\(\chi_{675}(661,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{11}{15}\right)\)