Properties

Label 2205.ej
Modulus $2205$
Conductor $2205$
Order $84$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,63,66]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(13,2205))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2205\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(8\) \(11\) \(13\) \(16\) \(17\) \(19\) \(22\) \(23\)
\(\chi_{2205}(13,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{11}{28}\right)\) \(1\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{65}{84}\right)\)
\(\chi_{2205}(202,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{84}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{53}{84}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{9}{28}\right)\) \(1\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{71}{84}\right)\)
\(\chi_{2205}(223,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{15}{28}\right)\) \(1\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{25}{84}\right)\)
\(\chi_{2205}(328,\cdot)\) \(1\) \(1\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{3}{28}\right)\) \(1\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{5}{84}\right)\)
\(\chi_{2205}(412,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{13}{28}\right)\) \(1\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{31}{84}\right)\)
\(\chi_{2205}(517,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{1}{28}\right)\) \(1\) \(e\left(\frac{53}{84}\right)\) \(e\left(\frac{11}{84}\right)\)
\(\chi_{2205}(643,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{23}{28}\right)\) \(1\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{29}{84}\right)\)
\(\chi_{2205}(727,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{5}{28}\right)\) \(1\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{55}{84}\right)\)
\(\chi_{2205}(853,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{27}{28}\right)\) \(1\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{73}{84}\right)\)
\(\chi_{2205}(958,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{15}{28}\right)\) \(1\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{53}{84}\right)\)
\(\chi_{2205}(1042,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{25}{28}\right)\) \(1\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{79}{84}\right)\)
\(\chi_{2205}(1147,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{13}{28}\right)\) \(1\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{59}{84}\right)\)
\(\chi_{2205}(1168,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{19}{28}\right)\) \(1\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{13}{84}\right)\)
\(\chi_{2205}(1357,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{17}{28}\right)\) \(1\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{19}{84}\right)\)
\(\chi_{2205}(1462,\cdot)\) \(1\) \(1\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{5}{28}\right)\) \(1\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{83}{84}\right)\)
\(\chi_{2205}(1483,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{11}{28}\right)\) \(1\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{37}{84}\right)\)
\(\chi_{2205}(1588,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{27}{28}\right)\) \(1\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{17}{84}\right)\)
\(\chi_{2205}(1672,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{84}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{25}{84}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{9}{28}\right)\) \(1\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{43}{84}\right)\)
\(\chi_{2205}(1777,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{25}{28}\right)\) \(1\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{23}{84}\right)\)
\(\chi_{2205}(1798,\cdot)\) \(1\) \(1\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{3}{28}\right)\) \(1\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{61}{84}\right)\)
\(\chi_{2205}(1903,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{19}{28}\right)\) \(1\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{41}{84}\right)\)
\(\chi_{2205}(1987,\cdot)\) \(1\) \(1\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{28}\right)\) \(1\) \(e\left(\frac{25}{84}\right)\) \(e\left(\frac{67}{84}\right)\)
\(\chi_{2205}(2092,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{17}{28}\right)\) \(1\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{47}{84}\right)\)
\(\chi_{2205}(2113,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{23}{28}\right)\) \(1\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{1}{84}\right)\)