Properties

Label 2009.ci
Modulus $2009$
Conductor $2009$
Order $420$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, base_ring=CyclotomicField(420))
 
M = H._module
 
chi = DirichletCharacter(H, M([290,231]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(5,2009))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

First 31 of 96 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(11\) \(12\)
\(\chi_{2009}(5,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{210}\right)\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{53}{105}\right)\) \(e\left(\frac{13}{105}\right)\) \(e\left(\frac{27}{140}\right)\) \(e\left(\frac{53}{70}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{79}{210}\right)\) \(e\left(\frac{113}{420}\right)\) \(e\left(\frac{187}{420}\right)\)
\(\chi_{2009}(33,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{210}\right)\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{22}{105}\right)\) \(e\left(\frac{113}{140}\right)\) \(e\left(\frac{17}{70}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{61}{210}\right)\) \(e\left(\frac{167}{420}\right)\) \(e\left(\frac{373}{420}\right)\)
\(\chi_{2009}(61,\cdot)\) \(-1\) \(1\) \(e\left(\frac{191}{210}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{86}{105}\right)\) \(e\left(\frac{31}{105}\right)\) \(e\left(\frac{129}{140}\right)\) \(e\left(\frac{51}{70}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{43}{210}\right)\) \(e\left(\frac{11}{420}\right)\) \(e\left(\frac{349}{420}\right)\)
\(\chi_{2009}(87,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{210}\right)\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{13}{105}\right)\) \(e\left(\frac{23}{105}\right)\) \(e\left(\frac{107}{140}\right)\) \(e\left(\frac{13}{70}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{59}{210}\right)\) \(e\left(\frac{313}{420}\right)\) \(e\left(\frac{347}{420}\right)\)
\(\chi_{2009}(103,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{210}\right)\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{11}{105}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{139}{140}\right)\) \(e\left(\frac{11}{70}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{163}{210}\right)\) \(e\left(\frac{281}{420}\right)\) \(e\left(\frac{19}{420}\right)\)
\(\chi_{2009}(115,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{210}\right)\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{37}{105}\right)\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{73}{140}\right)\) \(e\left(\frac{37}{70}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{71}{210}\right)\) \(e\left(\frac{67}{420}\right)\) \(e\left(\frac{293}{420}\right)\)
\(\chi_{2009}(131,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{210}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{22}{105}\right)\) \(e\left(\frac{43}{140}\right)\) \(e\left(\frac{17}{70}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{61}{210}\right)\) \(e\left(\frac{377}{420}\right)\) \(e\left(\frac{163}{420}\right)\)
\(\chi_{2009}(143,\cdot)\) \(-1\) \(1\) \(e\left(\frac{61}{210}\right)\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{61}{105}\right)\) \(e\left(\frac{11}{105}\right)\) \(e\left(\frac{109}{140}\right)\) \(e\left(\frac{61}{70}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{83}{210}\right)\) \(e\left(\frac{31}{420}\right)\) \(e\left(\frac{29}{420}\right)\)
\(\chi_{2009}(159,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{210}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{23}{105}\right)\) \(e\left(\frac{73}{105}\right)\) \(e\left(\frac{17}{140}\right)\) \(e\left(\frac{23}{70}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{169}{210}\right)\) \(e\left(\frac{263}{420}\right)\) \(e\left(\frac{97}{420}\right)\)
\(\chi_{2009}(185,\cdot)\) \(-1\) \(1\) \(e\left(\frac{181}{210}\right)\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{86}{105}\right)\) \(e\left(\frac{79}{140}\right)\) \(e\left(\frac{41}{70}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{143}{210}\right)\) \(e\left(\frac{61}{420}\right)\) \(e\left(\frac{179}{420}\right)\)
\(\chi_{2009}(213,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{210}\right)\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{37}{105}\right)\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{3}{140}\right)\) \(e\left(\frac{37}{70}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{71}{210}\right)\) \(e\left(\frac{277}{420}\right)\) \(e\left(\frac{83}{420}\right)\)
\(\chi_{2009}(241,\cdot)\) \(-1\) \(1\) \(e\left(\frac{103}{210}\right)\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{103}{105}\right)\) \(e\left(\frac{53}{105}\right)\) \(e\left(\frac{137}{140}\right)\) \(e\left(\frac{33}{70}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{209}{210}\right)\) \(e\left(\frac{283}{420}\right)\) \(e\left(\frac{197}{420}\right)\)
\(\chi_{2009}(248,\cdot)\) \(-1\) \(1\) \(e\left(\frac{109}{210}\right)\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{4}{105}\right)\) \(e\left(\frac{104}{105}\right)\) \(e\left(\frac{41}{140}\right)\) \(e\left(\frac{39}{70}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{107}{210}\right)\) \(e\left(\frac{379}{420}\right)\) \(e\left(\frac{341}{420}\right)\)
\(\chi_{2009}(285,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{210}\right)\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{29}{105}\right)\) \(e\left(\frac{19}{105}\right)\) \(e\left(\frac{131}{140}\right)\) \(e\left(\frac{29}{70}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{67}{210}\right)\) \(e\left(\frac{149}{420}\right)\) \(e\left(\frac{31}{420}\right)\)
\(\chi_{2009}(292,\cdot)\) \(-1\) \(1\) \(e\left(\frac{83}{210}\right)\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{83}{105}\right)\) \(e\left(\frac{58}{105}\right)\) \(e\left(\frac{107}{140}\right)\) \(e\left(\frac{13}{70}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{199}{210}\right)\) \(e\left(\frac{173}{420}\right)\) \(e\left(\frac{67}{420}\right)\)
\(\chi_{2009}(320,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{210}\right)\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{47}{105}\right)\) \(e\left(\frac{67}{105}\right)\) \(e\left(\frac{53}{140}\right)\) \(e\left(\frac{47}{70}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{181}{210}\right)\) \(e\left(\frac{227}{420}\right)\) \(e\left(\frac{253}{420}\right)\)
\(\chi_{2009}(348,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{210}\right)\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{11}{105}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{69}{140}\right)\) \(e\left(\frac{11}{70}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{163}{210}\right)\) \(e\left(\frac{71}{420}\right)\) \(e\left(\frac{229}{420}\right)\)
\(\chi_{2009}(367,\cdot)\) \(-1\) \(1\) \(e\left(\frac{169}{210}\right)\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{64}{105}\right)\) \(e\left(\frac{89}{105}\right)\) \(e\left(\frac{131}{140}\right)\) \(e\left(\frac{29}{70}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{137}{210}\right)\) \(e\left(\frac{289}{420}\right)\) \(e\left(\frac{311}{420}\right)\)
\(\chi_{2009}(390,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{210}\right)\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{41}{105}\right)\) \(e\left(\frac{16}{105}\right)\) \(e\left(\frac{79}{140}\right)\) \(e\left(\frac{41}{70}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{73}{210}\right)\) \(e\left(\frac{341}{420}\right)\) \(e\left(\frac{319}{420}\right)\)
\(\chi_{2009}(402,\cdot)\) \(-1\) \(1\) \(e\left(\frac{157}{210}\right)\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{52}{105}\right)\) \(e\left(\frac{92}{105}\right)\) \(e\left(\frac{113}{140}\right)\) \(e\left(\frac{17}{70}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{131}{210}\right)\) \(e\left(\frac{307}{420}\right)\) \(e\left(\frac{233}{420}\right)\)
\(\chi_{2009}(418,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{210}\right)\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{47}{105}\right)\) \(e\left(\frac{67}{105}\right)\) \(e\left(\frac{123}{140}\right)\) \(e\left(\frac{47}{70}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{181}{210}\right)\) \(e\left(\frac{17}{420}\right)\) \(e\left(\frac{43}{420}\right)\)
\(\chi_{2009}(430,\cdot)\) \(-1\) \(1\) \(e\left(\frac{181}{210}\right)\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{86}{105}\right)\) \(e\left(\frac{9}{140}\right)\) \(e\left(\frac{41}{70}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{143}{210}\right)\) \(e\left(\frac{271}{420}\right)\) \(e\left(\frac{389}{420}\right)\)
\(\chi_{2009}(446,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{210}\right)\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{53}{105}\right)\) \(e\left(\frac{13}{105}\right)\) \(e\left(\frac{97}{140}\right)\) \(e\left(\frac{53}{70}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{79}{210}\right)\) \(e\left(\frac{323}{420}\right)\) \(e\left(\frac{397}{420}\right)\)
\(\chi_{2009}(453,\cdot)\) \(-1\) \(1\) \(e\left(\frac{149}{210}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{44}{105}\right)\) \(e\left(\frac{94}{105}\right)\) \(e\left(\frac{101}{140}\right)\) \(e\left(\frac{9}{70}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{127}{210}\right)\) \(e\left(\frac{179}{420}\right)\) \(e\left(\frac{181}{420}\right)\)
\(\chi_{2009}(500,\cdot)\) \(-1\) \(1\) \(e\left(\frac{157}{210}\right)\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{52}{105}\right)\) \(e\left(\frac{92}{105}\right)\) \(e\left(\frac{43}{140}\right)\) \(e\left(\frac{17}{70}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{131}{210}\right)\) \(e\left(\frac{97}{420}\right)\) \(e\left(\frac{23}{420}\right)\)
\(\chi_{2009}(528,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{210}\right)\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{13}{105}\right)\) \(e\left(\frac{23}{105}\right)\) \(e\left(\frac{37}{140}\right)\) \(e\left(\frac{13}{70}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{59}{210}\right)\) \(e\left(\frac{103}{420}\right)\) \(e\left(\frac{137}{420}\right)\)
\(\chi_{2009}(535,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{210}\right)\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{19}{105}\right)\) \(e\left(\frac{74}{105}\right)\) \(e\left(\frac{81}{140}\right)\) \(e\left(\frac{19}{70}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{167}{210}\right)\) \(e\left(\frac{199}{420}\right)\) \(e\left(\frac{281}{420}\right)\)
\(\chi_{2009}(572,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{210}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{59}{105}\right)\) \(e\left(\frac{64}{105}\right)\) \(e\left(\frac{71}{140}\right)\) \(e\left(\frac{59}{70}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{187}{210}\right)\) \(e\left(\frac{209}{420}\right)\) \(e\left(\frac{331}{420}\right)\)
\(\chi_{2009}(579,\cdot)\) \(-1\) \(1\) \(e\left(\frac{113}{210}\right)\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{8}{105}\right)\) \(e\left(\frac{103}{105}\right)\) \(e\left(\frac{47}{140}\right)\) \(e\left(\frac{43}{70}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{109}{210}\right)\) \(e\left(\frac{233}{420}\right)\) \(e\left(\frac{367}{420}\right)\)
\(\chi_{2009}(635,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{210}\right)\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{41}{105}\right)\) \(e\left(\frac{16}{105}\right)\) \(e\left(\frac{9}{140}\right)\) \(e\left(\frac{41}{70}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{73}{210}\right)\) \(e\left(\frac{131}{420}\right)\) \(e\left(\frac{109}{420}\right)\)
\(\chi_{2009}(654,\cdot)\) \(-1\) \(1\) \(e\left(\frac{79}{210}\right)\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{79}{105}\right)\) \(e\left(\frac{59}{105}\right)\) \(e\left(\frac{31}{140}\right)\) \(e\left(\frac{9}{70}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{197}{210}\right)\) \(e\left(\frac{109}{420}\right)\) \(e\left(\frac{251}{420}\right)\)