Properties

Label 837.ca
Modulus $837$
Conductor $837$
Order $45$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(837, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([80,84]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(7,837))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(837\)
Conductor: \(837\)
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: 45.45.13198639294739821268975716162663306052858909500289219592060310551297322480112649057056185531676266039663602421928889.2

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{837}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{7}{45}\right)\)
\(\chi_{837}(40,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{8}{45}\right)\)
\(\chi_{837}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{14}{45}\right)\)
\(\chi_{837}(103,\cdot)\) \(1\) \(1\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{41}{45}\right)\)
\(\chi_{837}(169,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{43}{45}\right)\)
\(\chi_{837}(175,\cdot)\) \(1\) \(1\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{17}{45}\right)\)
\(\chi_{837}(205,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{31}{45}\right)\)
\(\chi_{837}(214,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{19}{45}\right)\)
\(\chi_{837}(286,\cdot)\) \(1\) \(1\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{22}{45}\right)\)
\(\chi_{837}(319,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{23}{45}\right)\)
\(\chi_{837}(328,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{29}{45}\right)\)
\(\chi_{837}(382,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{11}{45}\right)\)
\(\chi_{837}(448,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{13}{45}\right)\)
\(\chi_{837}(454,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{32}{45}\right)\)
\(\chi_{837}(484,\cdot)\) \(1\) \(1\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{1}{45}\right)\)
\(\chi_{837}(493,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{34}{45}\right)\)
\(\chi_{837}(565,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{37}{45}\right)\)
\(\chi_{837}(598,\cdot)\) \(1\) \(1\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{38}{45}\right)\)
\(\chi_{837}(607,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{44}{45}\right)\)
\(\chi_{837}(661,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{26}{45}\right)\)
\(\chi_{837}(727,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{28}{45}\right)\)
\(\chi_{837}(733,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{2}{45}\right)\)
\(\chi_{837}(763,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{16}{45}\right)\)
\(\chi_{837}(772,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{4}{45}\right)\)