Properties

Label 1113.bp
Modulus $1113$
Conductor $1113$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1113, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,13,15]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(17,1113))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1113\)
Conductor: \(1113\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(11\) \(13\) \(16\) \(17\) \(19\)
\(\chi_{1113}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{39}\right)\) \(e\left(\frac{2}{39}\right)\) \(e\left(\frac{29}{78}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{31}{78}\right)\) \(e\left(\frac{25}{78}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{4}{39}\right)\) \(e\left(\frac{23}{39}\right)\) \(e\left(\frac{37}{39}\right)\)
\(\chi_{1113}(38,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{39}\right)\) \(e\left(\frac{5}{39}\right)\) \(e\left(\frac{53}{78}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{19}{78}\right)\) \(e\left(\frac{43}{78}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{10}{39}\right)\) \(e\left(\frac{38}{39}\right)\) \(e\left(\frac{34}{39}\right)\)
\(\chi_{1113}(59,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{39}\right)\) \(e\left(\frac{14}{39}\right)\) \(e\left(\frac{47}{78}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{61}{78}\right)\) \(e\left(\frac{19}{78}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{28}{39}\right)\) \(e\left(\frac{5}{39}\right)\) \(e\left(\frac{25}{39}\right)\)
\(\chi_{1113}(110,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{39}\right)\) \(e\left(\frac{16}{39}\right)\) \(e\left(\frac{37}{78}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{53}{78}\right)\) \(e\left(\frac{5}{78}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{32}{39}\right)\) \(e\left(\frac{28}{39}\right)\) \(e\left(\frac{23}{39}\right)\)
\(\chi_{1113}(131,\cdot)\) \(1\) \(1\) \(e\left(\frac{38}{39}\right)\) \(e\left(\frac{37}{39}\right)\) \(e\left(\frac{49}{78}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{47}{78}\right)\) \(e\left(\frac{53}{78}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{35}{39}\right)\) \(e\left(\frac{16}{39}\right)\) \(e\left(\frac{2}{39}\right)\)
\(\chi_{1113}(143,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{39}\right)\) \(e\left(\frac{32}{39}\right)\) \(e\left(\frac{35}{78}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{67}{78}\right)\) \(e\left(\frac{49}{78}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{25}{39}\right)\) \(e\left(\frac{17}{39}\right)\) \(e\left(\frac{7}{39}\right)\)
\(\chi_{1113}(269,\cdot)\) \(1\) \(1\) \(e\left(\frac{34}{39}\right)\) \(e\left(\frac{29}{39}\right)\) \(e\left(\frac{11}{78}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{1}{78}\right)\) \(e\left(\frac{31}{78}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{19}{39}\right)\) \(e\left(\frac{2}{39}\right)\) \(e\left(\frac{10}{39}\right)\)
\(\chi_{1113}(290,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{39}\right)\) \(e\left(\frac{11}{39}\right)\) \(e\left(\frac{23}{78}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{73}{78}\right)\) \(e\left(\frac{1}{78}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{22}{39}\right)\) \(e\left(\frac{29}{39}\right)\) \(e\left(\frac{28}{39}\right)\)
\(\chi_{1113}(467,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{39}\right)\) \(e\left(\frac{7}{39}\right)\) \(e\left(\frac{43}{78}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{11}{78}\right)\) \(e\left(\frac{29}{78}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{14}{39}\right)\) \(e\left(\frac{22}{39}\right)\) \(e\left(\frac{32}{39}\right)\)
\(\chi_{1113}(488,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{39}\right)\) \(e\left(\frac{22}{39}\right)\) \(e\left(\frac{7}{78}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{29}{78}\right)\) \(e\left(\frac{41}{78}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{5}{39}\right)\) \(e\left(\frac{19}{39}\right)\) \(e\left(\frac{17}{39}\right)\)
\(\chi_{1113}(626,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{39}\right)\) \(e\left(\frac{20}{39}\right)\) \(e\left(\frac{17}{78}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{37}{78}\right)\) \(e\left(\frac{55}{78}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{1}{39}\right)\) \(e\left(\frac{35}{39}\right)\) \(e\left(\frac{19}{39}\right)\)
\(\chi_{1113}(647,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{39}\right)\) \(e\left(\frac{35}{39}\right)\) \(e\left(\frac{59}{78}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{55}{78}\right)\) \(e\left(\frac{67}{78}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{31}{39}\right)\) \(e\left(\frac{32}{39}\right)\) \(e\left(\frac{4}{39}\right)\)
\(\chi_{1113}(698,\cdot)\) \(1\) \(1\) \(e\left(\frac{32}{39}\right)\) \(e\left(\frac{25}{39}\right)\) \(e\left(\frac{31}{78}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{17}{78}\right)\) \(e\left(\frac{59}{78}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{11}{39}\right)\) \(e\left(\frac{34}{39}\right)\) \(e\left(\frac{14}{39}\right)\)
\(\chi_{1113}(782,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{39}\right)\) \(e\left(\frac{10}{39}\right)\) \(e\left(\frac{67}{78}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{77}{78}\right)\) \(e\left(\frac{47}{78}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{20}{39}\right)\) \(e\left(\frac{37}{39}\right)\) \(e\left(\frac{29}{39}\right)\)
\(\chi_{1113}(824,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{39}\right)\) \(e\left(\frac{4}{39}\right)\) \(e\left(\frac{19}{78}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{23}{78}\right)\) \(e\left(\frac{11}{78}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{8}{39}\right)\) \(e\left(\frac{7}{39}\right)\) \(e\left(\frac{35}{39}\right)\)
\(\chi_{1113}(857,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{39}\right)\) \(e\left(\frac{38}{39}\right)\) \(e\left(\frac{5}{78}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{43}{78}\right)\) \(e\left(\frac{7}{78}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{37}{39}\right)\) \(e\left(\frac{8}{39}\right)\) \(e\left(\frac{1}{39}\right)\)
\(\chi_{1113}(908,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{39}\right)\) \(e\left(\frac{34}{39}\right)\) \(e\left(\frac{25}{78}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{59}{78}\right)\) \(e\left(\frac{35}{78}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{29}{39}\right)\) \(e\left(\frac{1}{39}\right)\) \(e\left(\frac{5}{39}\right)\)
\(\chi_{1113}(941,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{39}\right)\) \(e\left(\frac{23}{39}\right)\) \(e\left(\frac{41}{78}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{25}{78}\right)\) \(e\left(\frac{73}{78}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{7}{39}\right)\) \(e\left(\frac{11}{39}\right)\) \(e\left(\frac{16}{39}\right)\)
\(\chi_{1113}(971,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{39}\right)\) \(e\left(\frac{28}{39}\right)\) \(e\left(\frac{55}{78}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{5}{78}\right)\) \(e\left(\frac{77}{78}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{17}{39}\right)\) \(e\left(\frac{10}{39}\right)\) \(e\left(\frac{11}{39}\right)\)
\(\chi_{1113}(983,\cdot)\) \(1\) \(1\) \(e\left(\frac{28}{39}\right)\) \(e\left(\frac{17}{39}\right)\) \(e\left(\frac{71}{78}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{49}{78}\right)\) \(e\left(\frac{37}{78}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{34}{39}\right)\) \(e\left(\frac{20}{39}\right)\) \(e\left(\frac{22}{39}\right)\)
\(\chi_{1113}(992,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{39}\right)\) \(e\left(\frac{31}{39}\right)\) \(e\left(\frac{1}{78}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{71}{78}\right)\) \(e\left(\frac{17}{78}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{23}{39}\right)\) \(e\left(\frac{25}{39}\right)\) \(e\left(\frac{8}{39}\right)\)
\(\chi_{1113}(1013,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{39}\right)\) \(e\left(\frac{1}{39}\right)\) \(e\left(\frac{73}{78}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{35}{78}\right)\) \(e\left(\frac{71}{78}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{2}{39}\right)\) \(e\left(\frac{31}{39}\right)\) \(e\left(\frac{38}{39}\right)\)
\(\chi_{1113}(1067,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{39}\right)\) \(e\left(\frac{8}{39}\right)\) \(e\left(\frac{77}{78}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{7}{78}\right)\) \(e\left(\frac{61}{78}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{16}{39}\right)\) \(e\left(\frac{14}{39}\right)\) \(e\left(\frac{31}{39}\right)\)
\(\chi_{1113}(1097,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{39}\right)\) \(e\left(\frac{19}{39}\right)\) \(e\left(\frac{61}{78}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{41}{78}\right)\) \(e\left(\frac{23}{78}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{38}{39}\right)\) \(e\left(\frac{4}{39}\right)\) \(e\left(\frac{20}{39}\right)\)