Properties

Label 2000.cg
Modulus $2000$
Conductor $2000$
Order $100$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2000, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,25,92]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(21,2000))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2000\)
Conductor: \(2000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

First 31 of 40 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\)
\(\chi_{2000}(21,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{57}{100}\right)\)
\(\chi_{2000}(61,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{51}{100}\right)\)
\(\chi_{2000}(141,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{97}{100}\right)\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{59}{100}\right)\)
\(\chi_{2000}(181,\cdot)\) \(1\) \(1\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{13}{100}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{73}{100}\right)\)
\(\chi_{2000}(221,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{41}{100}\right)\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{27}{100}\right)\)
\(\chi_{2000}(261,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{39}{100}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{21}{100}\right)\)
\(\chi_{2000}(341,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{29}{100}\right)\)
\(\chi_{2000}(381,\cdot)\) \(1\) \(1\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{37}{100}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{43}{100}\right)\)
\(\chi_{2000}(421,\cdot)\) \(1\) \(1\) \(e\left(\frac{99}{100}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{23}{100}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{51}{100}\right)\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{97}{100}\right)\)
\(\chi_{2000}(461,\cdot)\) \(1\) \(1\) \(e\left(\frac{97}{100}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{91}{100}\right)\)
\(\chi_{2000}(541,\cdot)\) \(1\) \(1\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{41}{100}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{23}{100}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{99}{100}\right)\)
\(\chi_{2000}(581,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{13}{100}\right)\)
\(\chi_{2000}(621,\cdot)\) \(1\) \(1\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{67}{100}\right)\)
\(\chi_{2000}(661,\cdot)\) \(1\) \(1\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{41}{100}\right)\) \(e\left(\frac{99}{100}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{97}{100}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{61}{100}\right)\)
\(\chi_{2000}(741,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{100}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{13}{100}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{69}{100}\right)\)
\(\chi_{2000}(781,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{23}{100}\right)\) \(e\left(\frac{97}{100}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{83}{100}\right)\)
\(\chi_{2000}(821,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{97}{100}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{37}{100}\right)\)
\(\chi_{2000}(861,\cdot)\) \(1\) \(1\) \(e\left(\frac{77}{100}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{29}{100}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{31}{100}\right)\)
\(\chi_{2000}(941,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{100}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{37}{100}\right)\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{39}{100}\right)\)
\(\chi_{2000}(981,\cdot)\) \(1\) \(1\) \(e\left(\frac{51}{100}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{99}{100}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{53}{100}\right)\)
\(\chi_{2000}(1021,\cdot)\) \(1\) \(1\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{13}{100}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{39}{100}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{7}{100}\right)\)
\(\chi_{2000}(1061,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{77}{100}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{1}{100}\right)\)
\(\chi_{2000}(1141,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{29}{100}\right)\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{9}{100}\right)\)
\(\chi_{2000}(1181,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{100}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{23}{100}\right)\)
\(\chi_{2000}(1221,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{37}{100}\right)\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{29}{100}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{77}{100}\right)\)
\(\chi_{2000}(1261,\cdot)\) \(1\) \(1\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{51}{100}\right)\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{71}{100}\right)\)
\(\chi_{2000}(1341,\cdot)\) \(1\) \(1\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{99}{100}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{79}{100}\right)\)
\(\chi_{2000}(1381,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{93}{100}\right)\)
\(\chi_{2000}(1421,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{47}{100}\right)\)
\(\chi_{2000}(1461,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{41}{100}\right)\)
\(\chi_{2000}(1541,\cdot)\) \(1\) \(1\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{49}{100}\right)\)