Properties

Label 81.g
Modulus $81$
Conductor $81$
Order $27$
Real no
Primitive yes
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(81, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([2])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(4, 81)) 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("81.4"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(81\)
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: \(27\)
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: yes
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: \(\Q(\zeta_{27})\)
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 27 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{81}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{4}{27}\right)\)
\(\chi_{81}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{27}\right)\)
\(\chi_{81}(13,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{81}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{8}{27}\right)\)
\(\chi_{81}(22,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{1}{27}\right)\)
\(\chi_{81}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{11}{27}\right)\)
\(\chi_{81}(31,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{13}{27}\right)\)
\(\chi_{81}(34,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{14}{27}\right)\)
\(\chi_{81}(40,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{81}(43,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{27}\right)\)
\(\chi_{81}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{10}{27}\right)\)
\(\chi_{81}(52,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{20}{27}\right)\)
\(\chi_{81}(58,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{22}{27}\right)\)
\(\chi_{81}(61,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{23}{27}\right)\)
\(\chi_{81}(67,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{27}\right)\)
\(\chi_{81}(70,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{26}{27}\right)\)
\(\chi_{81}(76,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{81}(79,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{2}{27}\right)\)