# Properties

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

# Related objects

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