Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13688, base_ring=CyclotomicField(406)) M = H._module chi = DirichletCharacter(H, M([203,0,348,322]))
 
Copy content gp:[g,chi] = znchar(Mod(1383, 13688))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13688.1383");
 

Basic properties

Modulus: \(13688\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6844\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(406\)
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 \(\chi_{6844}(1383,\cdot)\)
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: no
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 13688.cz

\(\chi_{13688}(7,\cdot)\) \(\chi_{13688}(199,\cdot)\) \(\chi_{13688}(223,\cdot)\) \(\chi_{13688}(239,\cdot)\) \(\chi_{13688}(255,\cdot)\) \(\chi_{13688}(343,\cdot)\) \(\chi_{13688}(487,\cdot)\) \(\chi_{13688}(567,\cdot)\) \(\chi_{13688}(951,\cdot)\) \(\chi_{13688}(1031,\cdot)\) \(\chi_{13688}(1039,\cdot)\) \(\chi_{13688}(1167,\cdot)\) \(\chi_{13688}(1183,\cdot)\) \(\chi_{13688}(1383,\cdot)\) \(\chi_{13688}(1495,\cdot)\) \(\chi_{13688}(1503,\cdot)\) \(\chi_{13688}(1591,\cdot)\) \(\chi_{13688}(1615,\cdot)\) \(\chi_{13688}(1727,\cdot)\) \(\chi_{13688}(1823,\cdot)\) \(\chi_{13688}(1959,\cdot)\) \(\chi_{13688}(1967,\cdot)\) \(\chi_{13688}(2055,\cdot)\) \(\chi_{13688}(2111,\cdot)\) \(\chi_{13688}(2199,\cdot)\) \(\chi_{13688}(2287,\cdot)\) \(\chi_{13688}(2327,\cdot)\) \(\chi_{13688}(2423,\cdot)\) \(\chi_{13688}(2431,\cdot)\) \(\chi_{13688}(2519,\cdot)\) ...

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: $\Q(\zeta_{203})$
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 406 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((3423,6845,5193,10209)\) → \((-1,1,e\left(\frac{6}{7}\right),e\left(\frac{23}{29}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 13688 }(1383, a) \) \(-1\)\(1\)\(e\left(\frac{179}{406}\right)\)\(e\left(\frac{125}{203}\right)\)\(e\left(\frac{25}{406}\right)\)\(e\left(\frac{179}{203}\right)\)\(e\left(\frac{307}{406}\right)\)\(e\left(\frac{24}{203}\right)\)\(e\left(\frac{23}{406}\right)\)\(e\left(\frac{21}{29}\right)\)\(e\left(\frac{143}{406}\right)\)\(e\left(\frac{102}{203}\right)\)
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)
 
\( \chi_{ 13688 }(1383,a) \;\) at \(\;a = \) e.g. 2