Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3330, base_ring=CyclotomicField(36)) M = H._module chi = DirichletCharacter(H, M([0,18,23]))
 
Copy content gp:[g,chi] = znchar(Mod(1189, 3330))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3330.1189");
 

Basic properties

Modulus: \(3330\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(185\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(36\)
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 \(\chi_{185}(79,\cdot)\)
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);
 

Galois orbit 3330.fs

\(\chi_{3330}(19,\cdot)\) \(\chi_{3330}(109,\cdot)\) \(\chi_{3330}(649,\cdot)\) \(\chi_{3330}(829,\cdot)\) \(\chi_{3330}(1189,\cdot)\) \(\chi_{3330}(1549,\cdot)\) \(\chi_{3330}(1909,\cdot)\) \(\chi_{3330}(2089,\cdot)\) \(\chi_{3330}(2629,\cdot)\) \(\chi_{3330}(2719,\cdot)\) \(\chi_{3330}(2899,\cdot)\) \(\chi_{3330}(3169,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
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: 36.0.29411719834995153896864925426307140281034671856927417346954345703125.1
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((371,667,631)\) → \((1,-1,e\left(\frac{23}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 3330 }(1189, a) \) \(-1\)\(1\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(-i\)\(e\left(\frac{5}{18}\right)\)\(-i\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 3330 }(1189,a) \;\) at \(\;a = \) e.g. 2