Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(72)) M = H._module chi = DirichletCharacter(H, M([0,45,10]))
 
Copy content gp:[g,chi] = znchar(Mod(1401, 2368))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2368.1401");
 

Basic properties

Modulus: \(2368\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1184\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(72\)
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 \(\chi_{1184}(661,\cdot)\)
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 2368.dv

\(\chi_{2368}(57,\cdot)\) \(\chi_{2368}(89,\cdot)\) \(\chi_{2368}(217,\cdot)\) \(\chi_{2368}(409,\cdot)\) \(\chi_{2368}(457,\cdot)\) \(\chi_{2368}(505,\cdot)\) \(\chi_{2368}(553,\cdot)\) \(\chi_{2368}(745,\cdot)\) \(\chi_{2368}(873,\cdot)\) \(\chi_{2368}(905,\cdot)\) \(\chi_{2368}(1017,\cdot)\) \(\chi_{2368}(1129,\cdot)\) \(\chi_{2368}(1241,\cdot)\) \(\chi_{2368}(1273,\cdot)\) \(\chi_{2368}(1401,\cdot)\) \(\chi_{2368}(1593,\cdot)\) \(\chi_{2368}(1641,\cdot)\) \(\chi_{2368}(1689,\cdot)\) \(\chi_{2368}(1737,\cdot)\) \(\chi_{2368}(1929,\cdot)\) \(\chi_{2368}(2057,\cdot)\) \(\chi_{2368}(2089,\cdot)\) \(\chi_{2368}(2201,\cdot)\) \(\chi_{2368}(2313,\cdot)\)

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: $\Q(\zeta_{72})$
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 72 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1407,1925,705)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{5}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2368 }(1401, a) \) \(-1\)\(1\)\(e\left(\frac{35}{72}\right)\)\(e\left(\frac{59}{72}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{65}{72}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{17}{72}\right)\)\(e\left(\frac{13}{72}\right)\)
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)
 
\( \chi_{ 2368 }(1401,a) \;\) at \(\;a = \) e.g. 2