Properties

Label 125.h
Modulus $125$
Conductor $125$
Order $50$
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(125, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(4,125))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(125\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{125}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{39}{50}\right)\)
\(\chi_{125}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{23}{50}\right)\)
\(\chi_{125}(14,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{27}{50}\right)\)
\(\chi_{125}(19,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{1}{50}\right)\)
\(\chi_{125}(29,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{9}{50}\right)\)
\(\chi_{125}(34,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{43}{50}\right)\)
\(\chi_{125}(39,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{47}{50}\right)\)
\(\chi_{125}(44,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{21}{50}\right)\)
\(\chi_{125}(54,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{29}{50}\right)\)
\(\chi_{125}(59,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{13}{50}\right)\)
\(\chi_{125}(64,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{17}{50}\right)\)
\(\chi_{125}(69,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{41}{50}\right)\)
\(\chi_{125}(79,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{49}{50}\right)\)
\(\chi_{125}(84,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{33}{50}\right)\)
\(\chi_{125}(89,\cdot)\) \(1\) \(1\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{37}{50}\right)\)
\(\chi_{125}(94,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{11}{50}\right)\)
\(\chi_{125}(104,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{19}{50}\right)\)
\(\chi_{125}(109,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{3}{50}\right)\)
\(\chi_{125}(114,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{7}{50}\right)\)
\(\chi_{125}(119,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{31}{50}\right)\)