Properties

Label 1019.d
Modulus $1019$
Conductor $1019$
Order $1018$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1019, base_ring=CyclotomicField(1018))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(2,1019))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1019\)
Conductor: \(1019\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1018\)
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: $\Q(\zeta_{509})$
Fixed field: Number field defined by a degree 1018 polynomial (not computed)

First 31 of 508 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{1019}(2,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{1018}\right)\) \(e\left(\frac{479}{509}\right)\) \(e\left(\frac{1}{509}\right)\) \(e\left(\frac{5}{509}\right)\) \(e\left(\frac{959}{1018}\right)\) \(e\left(\frac{363}{1018}\right)\) \(e\left(\frac{3}{1018}\right)\) \(e\left(\frac{449}{509}\right)\) \(e\left(\frac{11}{1018}\right)\) \(e\left(\frac{378}{509}\right)\)
\(\chi_{1019}(6,\cdot)\) \(-1\) \(1\) \(e\left(\frac{959}{1018}\right)\) \(e\left(\frac{243}{509}\right)\) \(e\left(\frac{450}{509}\right)\) \(e\left(\frac{214}{509}\right)\) \(e\left(\frac{427}{1018}\right)\) \(e\left(\frac{979}{1018}\right)\) \(e\left(\frac{841}{1018}\right)\) \(e\left(\frac{486}{509}\right)\) \(e\left(\frac{369}{1018}\right)\) \(e\left(\frac{94}{509}\right)\)
\(\chi_{1019}(7,\cdot)\) \(-1\) \(1\) \(e\left(\frac{363}{1018}\right)\) \(e\left(\frac{308}{509}\right)\) \(e\left(\frac{363}{509}\right)\) \(e\left(\frac{288}{509}\right)\) \(e\left(\frac{979}{1018}\right)\) \(e\left(\frac{447}{1018}\right)\) \(e\left(\frac{71}{1018}\right)\) \(e\left(\frac{107}{509}\right)\) \(e\left(\frac{939}{1018}\right)\) \(e\left(\frac{293}{509}\right)\)
\(\chi_{1019}(8,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{1018}\right)\) \(e\left(\frac{419}{509}\right)\) \(e\left(\frac{3}{509}\right)\) \(e\left(\frac{15}{509}\right)\) \(e\left(\frac{841}{1018}\right)\) \(e\left(\frac{71}{1018}\right)\) \(e\left(\frac{9}{1018}\right)\) \(e\left(\frac{329}{509}\right)\) \(e\left(\frac{33}{1018}\right)\) \(e\left(\frac{116}{509}\right)\)
\(\chi_{1019}(10,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{1018}\right)\) \(e\left(\frac{179}{509}\right)\) \(e\left(\frac{11}{509}\right)\) \(e\left(\frac{55}{509}\right)\) \(e\left(\frac{369}{1018}\right)\) \(e\left(\frac{939}{1018}\right)\) \(e\left(\frac{33}{1018}\right)\) \(e\left(\frac{358}{509}\right)\) \(e\left(\frac{121}{1018}\right)\) \(e\left(\frac{86}{509}\right)\)
\(\chi_{1019}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{289}{1018}\right)\) \(e\left(\frac{492}{509}\right)\) \(e\left(\frac{289}{509}\right)\) \(e\left(\frac{427}{509}\right)\) \(e\left(\frac{255}{1018}\right)\) \(e\left(\frac{53}{1018}\right)\) \(e\left(\frac{867}{1018}\right)\) \(e\left(\frac{475}{509}\right)\) \(e\left(\frac{125}{1018}\right)\) \(e\left(\frac{316}{509}\right)\)
\(\chi_{1019}(18,\cdot)\) \(-1\) \(1\) \(e\left(\frac{899}{1018}\right)\) \(e\left(\frac{7}{509}\right)\) \(e\left(\frac{390}{509}\right)\) \(e\left(\frac{423}{509}\right)\) \(e\left(\frac{913}{1018}\right)\) \(e\left(\frac{577}{1018}\right)\) \(e\left(\frac{661}{1018}\right)\) \(e\left(\frac{14}{509}\right)\) \(e\left(\frac{727}{1018}\right)\) \(e\left(\frac{319}{509}\right)\)
\(\chi_{1019}(21,\cdot)\) \(-1\) \(1\) \(e\left(\frac{303}{1018}\right)\) \(e\left(\frac{72}{509}\right)\) \(e\left(\frac{303}{509}\right)\) \(e\left(\frac{497}{509}\right)\) \(e\left(\frac{447}{1018}\right)\) \(e\left(\frac{45}{1018}\right)\) \(e\left(\frac{909}{1018}\right)\) \(e\left(\frac{144}{509}\right)\) \(e\left(\frac{279}{1018}\right)\) \(e\left(\frac{9}{509}\right)\)
\(\chi_{1019}(22,\cdot)\) \(-1\) \(1\) \(e\left(\frac{757}{1018}\right)\) \(e\left(\frac{195}{509}\right)\) \(e\left(\frac{248}{509}\right)\) \(e\left(\frac{222}{509}\right)\) \(e\left(\frac{129}{1018}\right)\) \(e\left(\frac{949}{1018}\right)\) \(e\left(\frac{235}{1018}\right)\) \(e\left(\frac{390}{509}\right)\) \(e\left(\frac{183}{1018}\right)\) \(e\left(\frac{88}{509}\right)\)
\(\chi_{1019}(24,\cdot)\) \(-1\) \(1\) \(e\left(\frac{961}{1018}\right)\) \(e\left(\frac{183}{509}\right)\) \(e\left(\frac{452}{509}\right)\) \(e\left(\frac{224}{509}\right)\) \(e\left(\frac{309}{1018}\right)\) \(e\left(\frac{687}{1018}\right)\) \(e\left(\frac{847}{1018}\right)\) \(e\left(\frac{366}{509}\right)\) \(e\left(\frac{391}{1018}\right)\) \(e\left(\frac{341}{509}\right)\)
\(\chi_{1019}(28,\cdot)\) \(-1\) \(1\) \(e\left(\frac{365}{1018}\right)\) \(e\left(\frac{248}{509}\right)\) \(e\left(\frac{365}{509}\right)\) \(e\left(\frac{298}{509}\right)\) \(e\left(\frac{861}{1018}\right)\) \(e\left(\frac{155}{1018}\right)\) \(e\left(\frac{77}{1018}\right)\) \(e\left(\frac{496}{509}\right)\) \(e\left(\frac{961}{1018}\right)\) \(e\left(\frac{31}{509}\right)\)
\(\chi_{1019}(30,\cdot)\) \(-1\) \(1\) \(e\left(\frac{969}{1018}\right)\) \(e\left(\frac{452}{509}\right)\) \(e\left(\frac{460}{509}\right)\) \(e\left(\frac{264}{509}\right)\) \(e\left(\frac{855}{1018}\right)\) \(e\left(\frac{537}{1018}\right)\) \(e\left(\frac{871}{1018}\right)\) \(e\left(\frac{395}{509}\right)\) \(e\left(\frac{479}{1018}\right)\) \(e\left(\frac{311}{509}\right)\)
\(\chi_{1019}(32,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{1018}\right)\) \(e\left(\frac{359}{509}\right)\) \(e\left(\frac{5}{509}\right)\) \(e\left(\frac{25}{509}\right)\) \(e\left(\frac{723}{1018}\right)\) \(e\left(\frac{797}{1018}\right)\) \(e\left(\frac{15}{1018}\right)\) \(e\left(\frac{209}{509}\right)\) \(e\left(\frac{55}{1018}\right)\) \(e\left(\frac{363}{509}\right)\)
\(\chi_{1019}(34,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{1018}\right)\) \(e\left(\frac{57}{509}\right)\) \(e\left(\frac{49}{509}\right)\) \(e\left(\frac{245}{509}\right)\) \(e\left(\frac{163}{1018}\right)\) \(e\left(\frac{481}{1018}\right)\) \(e\left(\frac{147}{1018}\right)\) \(e\left(\frac{114}{509}\right)\) \(e\left(\frac{539}{1018}\right)\) \(e\left(\frac{198}{509}\right)\)
\(\chi_{1019}(35,\cdot)\) \(-1\) \(1\) \(e\left(\frac{373}{1018}\right)\) \(e\left(\frac{8}{509}\right)\) \(e\left(\frac{373}{509}\right)\) \(e\left(\frac{338}{509}\right)\) \(e\left(\frac{389}{1018}\right)\) \(e\left(\frac{5}{1018}\right)\) \(e\left(\frac{101}{1018}\right)\) \(e\left(\frac{16}{509}\right)\) \(e\left(\frac{31}{1018}\right)\) \(e\left(\frac{1}{509}\right)\)
\(\chi_{1019}(37,\cdot)\) \(-1\) \(1\) \(e\left(\frac{701}{1018}\right)\) \(e\left(\frac{348}{509}\right)\) \(e\left(\frac{192}{509}\right)\) \(e\left(\frac{451}{509}\right)\) \(e\left(\frac{379}{1018}\right)\) \(e\left(\frac{981}{1018}\right)\) \(e\left(\frac{67}{1018}\right)\) \(e\left(\frac{187}{509}\right)\) \(e\left(\frac{585}{1018}\right)\) \(e\left(\frac{298}{509}\right)\)
\(\chi_{1019}(38,\cdot)\) \(-1\) \(1\) \(e\left(\frac{543}{1018}\right)\) \(e\left(\frac{507}{509}\right)\) \(e\left(\frac{34}{509}\right)\) \(e\left(\frac{170}{509}\right)\) \(e\left(\frac{539}{1018}\right)\) \(e\left(\frac{635}{1018}\right)\) \(e\left(\frac{611}{1018}\right)\) \(e\left(\frac{505}{509}\right)\) \(e\left(\frac{883}{1018}\right)\) \(e\left(\frac{127}{509}\right)\)
\(\chi_{1019}(39,\cdot)\) \(-1\) \(1\) \(e\left(\frac{229}{1018}\right)\) \(e\left(\frac{256}{509}\right)\) \(e\left(\frac{229}{509}\right)\) \(e\left(\frac{127}{509}\right)\) \(e\left(\frac{741}{1018}\right)\) \(e\left(\frac{669}{1018}\right)\) \(e\left(\frac{687}{1018}\right)\) \(e\left(\frac{3}{509}\right)\) \(e\left(\frac{483}{1018}\right)\) \(e\left(\frac{32}{509}\right)\)
\(\chi_{1019}(40,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{1018}\right)\) \(e\left(\frac{119}{509}\right)\) \(e\left(\frac{13}{509}\right)\) \(e\left(\frac{65}{509}\right)\) \(e\left(\frac{251}{1018}\right)\) \(e\left(\frac{647}{1018}\right)\) \(e\left(\frac{39}{1018}\right)\) \(e\left(\frac{238}{509}\right)\) \(e\left(\frac{143}{1018}\right)\) \(e\left(\frac{333}{509}\right)\)
\(\chi_{1019}(41,\cdot)\) \(-1\) \(1\) \(e\left(\frac{939}{1018}\right)\) \(e\left(\frac{334}{509}\right)\) \(e\left(\frac{430}{509}\right)\) \(e\left(\frac{114}{509}\right)\) \(e\left(\frac{589}{1018}\right)\) \(e\left(\frac{845}{1018}\right)\) \(e\left(\frac{781}{1018}\right)\) \(e\left(\frac{159}{509}\right)\) \(e\left(\frac{149}{1018}\right)\) \(e\left(\frac{169}{509}\right)\)
\(\chi_{1019}(46,\cdot)\) \(-1\) \(1\) \(e\left(\frac{115}{1018}\right)\) \(e\left(\frac{113}{509}\right)\) \(e\left(\frac{115}{509}\right)\) \(e\left(\frac{66}{509}\right)\) \(e\left(\frac{341}{1018}\right)\) \(e\left(\frac{7}{1018}\right)\) \(e\left(\frac{345}{1018}\right)\) \(e\left(\frac{226}{509}\right)\) \(e\left(\frac{247}{1018}\right)\) \(e\left(\frac{205}{509}\right)\)
\(\chi_{1019}(47,\cdot)\) \(-1\) \(1\) \(e\left(\frac{211}{1018}\right)\) \(e\left(\frac{287}{509}\right)\) \(e\left(\frac{211}{509}\right)\) \(e\left(\frac{37}{509}\right)\) \(e\left(\frac{785}{1018}\right)\) \(e\left(\frac{243}{1018}\right)\) \(e\left(\frac{633}{1018}\right)\) \(e\left(\frac{65}{509}\right)\) \(e\left(\frac{285}{1018}\right)\) \(e\left(\frac{354}{509}\right)\)
\(\chi_{1019}(50,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{1018}\right)\) \(e\left(\frac{388}{509}\right)\) \(e\left(\frac{21}{509}\right)\) \(e\left(\frac{105}{509}\right)\) \(e\left(\frac{797}{1018}\right)\) \(e\left(\frac{497}{1018}\right)\) \(e\left(\frac{63}{1018}\right)\) \(e\left(\frac{267}{509}\right)\) \(e\left(\frac{231}{1018}\right)\) \(e\left(\frac{303}{509}\right)\)
\(\chi_{1019}(52,\cdot)\) \(-1\) \(1\) \(e\left(\frac{291}{1018}\right)\) \(e\left(\frac{432}{509}\right)\) \(e\left(\frac{291}{509}\right)\) \(e\left(\frac{437}{509}\right)\) \(e\left(\frac{137}{1018}\right)\) \(e\left(\frac{779}{1018}\right)\) \(e\left(\frac{873}{1018}\right)\) \(e\left(\frac{355}{509}\right)\) \(e\left(\frac{147}{1018}\right)\) \(e\left(\frac{54}{509}\right)\)
\(\chi_{1019}(53,\cdot)\) \(-1\) \(1\) \(e\left(\frac{927}{1018}\right)\) \(e\left(\frac{185}{509}\right)\) \(e\left(\frac{418}{509}\right)\) \(e\left(\frac{54}{509}\right)\) \(e\left(\frac{279}{1018}\right)\) \(e\left(\frac{561}{1018}\right)\) \(e\left(\frac{745}{1018}\right)\) \(e\left(\frac{370}{509}\right)\) \(e\left(\frac{17}{1018}\right)\) \(e\left(\frac{214}{509}\right)\)
\(\chi_{1019}(54,\cdot)\) \(-1\) \(1\) \(e\left(\frac{839}{1018}\right)\) \(e\left(\frac{280}{509}\right)\) \(e\left(\frac{330}{509}\right)\) \(e\left(\frac{123}{509}\right)\) \(e\left(\frac{381}{1018}\right)\) \(e\left(\frac{175}{1018}\right)\) \(e\left(\frac{481}{1018}\right)\) \(e\left(\frac{51}{509}\right)\) \(e\left(\frac{67}{1018}\right)\) \(e\left(\frac{35}{509}\right)\)
\(\chi_{1019}(58,\cdot)\) \(-1\) \(1\) \(e\left(\frac{139}{1018}\right)\) \(e\left(\frac{411}{509}\right)\) \(e\left(\frac{139}{509}\right)\) \(e\left(\frac{186}{509}\right)\) \(e\left(\frac{961}{1018}\right)\) \(e\left(\frac{575}{1018}\right)\) \(e\left(\frac{417}{1018}\right)\) \(e\left(\frac{313}{509}\right)\) \(e\left(\frac{511}{1018}\right)\) \(e\left(\frac{115}{509}\right)\)
\(\chi_{1019}(59,\cdot)\) \(-1\) \(1\) \(e\left(\frac{465}{1018}\right)\) \(e\left(\frac{302}{509}\right)\) \(e\left(\frac{465}{509}\right)\) \(e\left(\frac{289}{509}\right)\) \(e\left(\frac{51}{1018}\right)\) \(e\left(\frac{825}{1018}\right)\) \(e\left(\frac{377}{1018}\right)\) \(e\left(\frac{95}{509}\right)\) \(e\left(\frac{25}{1018}\right)\) \(e\left(\frac{165}{509}\right)\)
\(\chi_{1019}(61,\cdot)\) \(-1\) \(1\) \(e\left(\frac{851}{1018}\right)\) \(e\left(\frac{429}{509}\right)\) \(e\left(\frac{342}{509}\right)\) \(e\left(\frac{183}{509}\right)\) \(e\left(\frac{691}{1018}\right)\) \(e\left(\frac{459}{1018}\right)\) \(e\left(\frac{517}{1018}\right)\) \(e\left(\frac{349}{509}\right)\) \(e\left(\frac{199}{1018}\right)\) \(e\left(\frac{499}{509}\right)\)
\(\chi_{1019}(62,\cdot)\) \(-1\) \(1\) \(e\left(\frac{325}{1018}\right)\) \(e\left(\frac{430}{509}\right)\) \(e\left(\frac{325}{509}\right)\) \(e\left(\frac{98}{509}\right)\) \(e\left(\frac{167}{1018}\right)\) \(e\left(\frac{905}{1018}\right)\) \(e\left(\frac{975}{1018}\right)\) \(e\left(\frac{351}{509}\right)\) \(e\left(\frac{521}{1018}\right)\) \(e\left(\frac{181}{509}\right)\)
\(\chi_{1019}(63,\cdot)\) \(-1\) \(1\) \(e\left(\frac{243}{1018}\right)\) \(e\left(\frac{345}{509}\right)\) \(e\left(\frac{243}{509}\right)\) \(e\left(\frac{197}{509}\right)\) \(e\left(\frac{933}{1018}\right)\) \(e\left(\frac{661}{1018}\right)\) \(e\left(\frac{729}{1018}\right)\) \(e\left(\frac{181}{509}\right)\) \(e\left(\frac{637}{1018}\right)\) \(e\left(\frac{234}{509}\right)\)