# Properties

 Label 500.m Modulus $500$ Conductor $125$ Order $25$ Real no Primitive no 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(50))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,46]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(21,500))

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Basic properties

 Modulus: $$500$$ Conductor: $$125$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$25$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 125.g 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: $$\Q(\zeta_{25})$$ Fixed field: 25.25.338813178901720135627329000271856784820556640625.1

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$
$$\chi_{500}(21,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$
$$\chi_{500}(41,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$
$$\chi_{500}(61,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$
$$\chi_{500}(81,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$
$$\chi_{500}(121,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$
$$\chi_{500}(141,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$
$$\chi_{500}(161,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$
$$\chi_{500}(181,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$
$$\chi_{500}(221,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$
$$\chi_{500}(241,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$
$$\chi_{500}(261,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$
$$\chi_{500}(281,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$
$$\chi_{500}(321,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$
$$\chi_{500}(341,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$
$$\chi_{500}(361,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$
$$\chi_{500}(381,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$
$$\chi_{500}(421,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$
$$\chi_{500}(441,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$
$$\chi_{500}(461,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$
$$\chi_{500}(481,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$