Properties

Label 2205.ee
Modulus $2205$
Conductor $735$
Order $84$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2205, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([42,63,20]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(53,2205))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2205\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 735.bt
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 \(-1\) \(1\) \(2\) \(4\) \(8\) \(11\) \(13\) \(16\) \(17\) \(19\) \(22\) \(23\)
\(\chi_{2205}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{67}{84}\right)\)
\(\chi_{2205}(107,\cdot)\) \(1\) \(1\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{5}{84}\right)\)
\(\chi_{2205}(233,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{25}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{59}{84}\right)\)
\(\chi_{2205}(242,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{13}{84}\right)\)
\(\chi_{2205}(368,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{19}{84}\right)\)
\(\chi_{2205}(548,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{47}{84}\right)\)
\(\chi_{2205}(683,\cdot)\) \(1\) \(1\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{55}{84}\right)\)
\(\chi_{2205}(737,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{65}{84}\right)\)
\(\chi_{2205}(872,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{1}{84}\right)\)
\(\chi_{2205}(1052,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{53}{84}\right)\)
\(\chi_{2205}(1178,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{23}{84}\right)\)
\(\chi_{2205}(1187,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{37}{84}\right)\)
\(\chi_{2205}(1313,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{84}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{43}{84}\right)\)
\(\chi_{2205}(1367,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{41}{84}\right)\)
\(\chi_{2205}(1493,\cdot)\) \(1\) \(1\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{11}{84}\right)\)
\(\chi_{2205}(1502,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{73}{84}\right)\)
\(\chi_{2205}(1628,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{79}{84}\right)\)
\(\chi_{2205}(1682,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{29}{84}\right)\)
\(\chi_{2205}(1808,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{83}{84}\right)\)
\(\chi_{2205}(1817,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{25}{84}\right)\)
\(\chi_{2205}(1943,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{53}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{31}{84}\right)\)
\(\chi_{2205}(1997,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{17}{84}\right)\)
\(\chi_{2205}(2123,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{84}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{71}{84}\right)\)
\(\chi_{2205}(2132,\cdot)\) \(1\) \(1\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{61}{84}\right)\)