Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([50,51,20]))
 
Copy content gp:[g,chi] = znchar(Mod(1472, 1575))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.1472");
 

Basic properties

Modulus: \(1575\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1575\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(60\)
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);
 

Galois orbit 1575.em

\(\chi_{1575}(23,\cdot)\) \(\chi_{1575}(137,\cdot)\) \(\chi_{1575}(212,\cdot)\) \(\chi_{1575}(263,\cdot)\) \(\chi_{1575}(338,\cdot)\) \(\chi_{1575}(452,\cdot)\) \(\chi_{1575}(527,\cdot)\) \(\chi_{1575}(578,\cdot)\) \(\chi_{1575}(653,\cdot)\) \(\chi_{1575}(767,\cdot)\) \(\chi_{1575}(842,\cdot)\) \(\chi_{1575}(1208,\cdot)\) \(\chi_{1575}(1283,\cdot)\) \(\chi_{1575}(1397,\cdot)\) \(\chi_{1575}(1472,\cdot)\) \(\chi_{1575}(1523,\cdot)\)

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: \(\Q(\zeta_{60})\)
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 60 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1226,127,451)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{17}{20}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 1575 }(1472, a) \) \(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{11}{60}\right)\)
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)
 
\( \chi_{ 1575 }(1472,a) \;\) at \(\;a = \) e.g. 2