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Properties

Label 1000.be
Modulus $1000$
Conductor $125$
Order $50$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(50)) M = H._module chi = DirichletCharacter(H, M([0,0,7])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(9, 1000)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1000.9"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(1000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(125\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(50\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from 125.h
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Related number fields

Field of values: \(\Q(\zeta_{25})\)
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 50 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\)
\(\chi_{1000}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{47}{50}\right)\)
\(\chi_{1000}(89,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{43}{50}\right)\)
\(\chi_{1000}(129,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{21}{50}\right)\)
\(\chi_{1000}(169,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{19}{50}\right)\)
\(\chi_{1000}(209,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{37}{50}\right)\)
\(\chi_{1000}(289,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{33}{50}\right)\)
\(\chi_{1000}(329,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{11}{50}\right)\)
\(\chi_{1000}(369,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{9}{50}\right)\)
\(\chi_{1000}(409,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{27}{50}\right)\)
\(\chi_{1000}(489,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{23}{50}\right)\)
\(\chi_{1000}(529,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{1}{50}\right)\)
\(\chi_{1000}(569,\cdot)\) \(1\) \(1\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{49}{50}\right)\)
\(\chi_{1000}(609,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{17}{50}\right)\)
\(\chi_{1000}(689,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{13}{50}\right)\)
\(\chi_{1000}(729,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{41}{50}\right)\)
\(\chi_{1000}(769,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{39}{50}\right)\)
\(\chi_{1000}(809,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{7}{50}\right)\)
\(\chi_{1000}(889,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{3}{50}\right)\)
\(\chi_{1000}(929,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{31}{50}\right)\)
\(\chi_{1000}(969,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{29}{50}\right)\)