Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20181, base_ring=CyclotomicField(186)) M = H._module chi = DirichletCharacter(H, M([0,62,16]))
 
Copy content gp:[g,chi] = znchar(Mod(7786, 20181))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20181.7786");
 

Basic properties

Modulus: \(20181\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6727\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(93\)
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 \(\chi_{6727}(1059,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 20181.dl

\(\chi_{20181}(25,\cdot)\) \(\chi_{20181}(625,\cdot)\) \(\chi_{20181}(676,\cdot)\) \(\chi_{20181}(1276,\cdot)\) \(\chi_{20181}(1327,\cdot)\) \(\chi_{20181}(1927,\cdot)\) \(\chi_{20181}(1978,\cdot)\) \(\chi_{20181}(2578,\cdot)\) \(\chi_{20181}(2629,\cdot)\) \(\chi_{20181}(3229,\cdot)\) \(\chi_{20181}(3280,\cdot)\) \(\chi_{20181}(3880,\cdot)\) \(\chi_{20181}(3931,\cdot)\) \(\chi_{20181}(4531,\cdot)\) \(\chi_{20181}(4582,\cdot)\) \(\chi_{20181}(5182,\cdot)\) \(\chi_{20181}(5233,\cdot)\) \(\chi_{20181}(5833,\cdot)\) \(\chi_{20181}(5884,\cdot)\) \(\chi_{20181}(6484,\cdot)\) \(\chi_{20181}(6535,\cdot)\) \(\chi_{20181}(7135,\cdot)\) \(\chi_{20181}(7186,\cdot)\) \(\chi_{20181}(7786,\cdot)\) \(\chi_{20181}(7837,\cdot)\) \(\chi_{20181}(8437,\cdot)\) \(\chi_{20181}(8488,\cdot)\) \(\chi_{20181}(9139,\cdot)\) \(\chi_{20181}(9739,\cdot)\) \(\chi_{20181}(9790,\cdot)\) ...

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: $\Q(\zeta_{93})$
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 93 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((6728,14416,12496)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{8}{93}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 20181 }(7786, a) \) \(1\)\(1\)\(e\left(\frac{32}{93}\right)\)\(e\left(\frac{64}{93}\right)\)\(e\left(\frac{16}{31}\right)\)\(e\left(\frac{1}{31}\right)\)\(e\left(\frac{80}{93}\right)\)\(e\left(\frac{11}{93}\right)\)\(e\left(\frac{46}{93}\right)\)\(e\left(\frac{35}{93}\right)\)\(e\left(\frac{27}{31}\right)\)\(e\left(\frac{7}{93}\right)\)
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)
 
\( \chi_{ 20181 }(7786,a) \;\) at \(\;a = \) e.g. 2