Properties

Label 500.r
Modulus $500$
Conductor $500$
Order $100$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(500, base_ring=CyclotomicField(100))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([50,7]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(3,500))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(500\)
Conductor: \(500\)
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 \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\)
\(\chi_{500}(3,\cdot)\) \(1\) \(1\) \(e\left(\frac{99}{100}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{97}{100}\right)\)
\(\chi_{500}(23,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{1}{100}\right)\)
\(\chi_{500}(27,\cdot)\) \(1\) \(1\) \(e\left(\frac{97}{100}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{91}{100}\right)\)
\(\chi_{500}(47,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{100}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{87}{100}\right)\)
\(\chi_{500}(63,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{29}{100}\right)\)
\(\chi_{500}(67,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{100}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{23}{100}\right)\)
\(\chi_{500}(83,\cdot)\) \(1\) \(1\) \(e\left(\frac{51}{100}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{77}{100}\right)\) \(e\left(\frac{39}{100}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{53}{100}\right)\)
\(\chi_{500}(87,\cdot)\) \(1\) \(1\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{37}{100}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{99}{100}\right)\)
\(\chi_{500}(103,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{100}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{17}{100}\right)\)
\(\chi_{500}(123,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{23}{100}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{21}{100}\right)\)
\(\chi_{500}(127,\cdot)\) \(1\) \(1\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{39}{100}\right)\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{71}{100}\right)\)
\(\chi_{500}(147,\cdot)\) \(1\) \(1\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{37}{100}\right)\) \(e\left(\frac{67}{100}\right)\)
\(\chi_{500}(163,\cdot)\) \(1\) \(1\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{41}{100}\right)\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{39}{100}\right)\) \(e\left(\frac{49}{100}\right)\)
\(\chi_{500}(167,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{3}{100}\right)\)
\(\chi_{500}(183,\cdot)\) \(1\) \(1\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{99}{100}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{73}{100}\right)\)
\(\chi_{500}(187,\cdot)\) \(1\) \(1\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{77}{100}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{79}{100}\right)\)
\(\chi_{500}(203,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{37}{100}\right)\)
\(\chi_{500}(223,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{51}{100}\right)\) \(e\left(\frac{41}{100}\right)\)
\(\chi_{500}(227,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{13}{100}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{51}{100}\right)\)
\(\chi_{500}(247,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{23}{100}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{47}{100}\right)\)
\(\chi_{500}(263,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{100}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{69}{100}\right)\)
\(\chi_{500}(267,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{29}{100}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{13}{100}\right)\) \(e\left(\frac{83}{100}\right)\)
\(\chi_{500}(283,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{37}{100}\right)\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{23}{100}\right)\) \(e\left(\frac{93}{100}\right)\)
\(\chi_{500}(287,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{59}{100}\right)\)
\(\chi_{500}(303,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{13}{100}\right)\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{57}{100}\right)\)
\(\chi_{500}(323,\cdot)\) \(1\) \(1\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{61}{100}\right)\)
\(\chi_{500}(327,\cdot)\) \(1\) \(1\) \(e\left(\frac{77}{100}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{41}{100}\right)\) \(e\left(\frac{31}{100}\right)\)
\(\chi_{500}(347,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{97}{100}\right)\) \(e\left(\frac{27}{100}\right)\)
\(\chi_{500}(363,\cdot)\) \(1\) \(1\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{89}{100}\right)\)
\(\chi_{500}(367,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{63}{100}\right)\)
\(\chi_{500}(383,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{13}{100}\right)\)