Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1011, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([0,67]))
 
Copy content gp:[g,chi] = znchar(Mod(772, 1011))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1011.772");
 

Basic properties

Modulus: \(1011\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(337\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 \(\chi_{337}(98,\cdot)\)
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 1011.bh

\(\chi_{1011}(37,\cdot)\) \(\chi_{1011}(82,\cdot)\) \(\chi_{1011}(193,\cdot)\) \(\chi_{1011}(196,\cdot)\) \(\chi_{1011}(262,\cdot)\) \(\chi_{1011}(319,\cdot)\) \(\chi_{1011}(328,\cdot)\) \(\chi_{1011}(346,\cdot)\) \(\chi_{1011}(355,\cdot)\) \(\chi_{1011}(412,\cdot)\) \(\chi_{1011}(478,\cdot)\) \(\chi_{1011}(481,\cdot)\) \(\chi_{1011}(592,\cdot)\) \(\chi_{1011}(637,\cdot)\) \(\chi_{1011}(715,\cdot)\) \(\chi_{1011}(748,\cdot)\) \(\chi_{1011}(772,\cdot)\) \(\chi_{1011}(784,\cdot)\) \(\chi_{1011}(805,\cdot)\) \(\chi_{1011}(880,\cdot)\) \(\chi_{1011}(901,\cdot)\) \(\chi_{1011}(913,\cdot)\) \(\chi_{1011}(937,\cdot)\) \(\chi_{1011}(970,\cdot)\)

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: $\Q(\zeta_{84})$
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 84 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((338,10)\) → \((1,e\left(\frac{67}{84}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1011 }(772, a) \) \(1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{19}{21}\right)\)
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)
 
\( \chi_{ 1011 }(772,a) \;\) at \(\;a = \) e.g. 2