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(324, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([0,23])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(5, 324)) 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("324.5"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(324\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(81\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(54\)
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 81.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: odd
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: \(\Q(\zeta_{27})\)
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 54 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{324}(5,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{14}{27}\right)\)
\(\chi_{324}(29,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{324}(41,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{17}{27}\right)\)
\(\chi_{324}(65,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{13}{27}\right)\)
\(\chi_{324}(77,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{20}{27}\right)\)
\(\chi_{324}(101,\cdot)\) \(-1\) \(1\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{7}{27}\right)\)
\(\chi_{324}(113,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{23}{27}\right)\)
\(\chi_{324}(137,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{1}{27}\right)\)
\(\chi_{324}(149,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{26}{27}\right)\)
\(\chi_{324}(173,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{22}{27}\right)\)
\(\chi_{324}(185,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{2}{27}\right)\)
\(\chi_{324}(209,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{324}(221,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{5}{27}\right)\)
\(\chi_{324}(245,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{10}{27}\right)\)
\(\chi_{324}(257,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{8}{27}\right)\)
\(\chi_{324}(281,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{4}{27}\right)\)
\(\chi_{324}(293,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{11}{27}\right)\)
\(\chi_{324}(317,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{25}{27}\right)\)