Properties

Label 500.m
Modulus $500$
Conductor $125$
Order $25$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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