↑

Properties

Label 91091.20
Modulus $91091$
Conductor $91091$
Order $5460$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91091, base_ring=CyclotomicField(5460)) M = H._module chi = DirichletCharacter(H, M([5070,3276,385]))
 
Copy content gp:[g,chi] = znchar(Mod(20, 91091))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("91091.20");
 

Basic properties

Modulus: \(91091\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(91091\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(5460\)
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 91091.rz

\(\chi_{91091}(20,\cdot)\) \(\chi_{91091}(202,\cdot)\) \(\chi_{91091}(223,\cdot)\) \(\chi_{91091}(279,\cdot)\) \(\chi_{91091}(405,\cdot)\) \(\chi_{91091}(531,\cdot)\) \(\chi_{91091}(566,\cdot)\) \(\chi_{91091}(643,\cdot)\) \(\chi_{91091}(713,\cdot)\) \(\chi_{91091}(839,\cdot)\) \(\chi_{91091}(895,\cdot)\) \(\chi_{91091}(916,\cdot)\) \(\chi_{91091}(951,\cdot)\) \(\chi_{91091}(1021,\cdot)\) \(\chi_{91091}(1098,\cdot)\) \(\chi_{91091}(1203,\cdot)\) \(\chi_{91091}(1259,\cdot)\) \(\chi_{91091}(1280,\cdot)\) \(\chi_{91091}(1406,\cdot)\) \(\chi_{91091}(1532,\cdot)\)

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: $\Q(\zeta_{5460})$
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 5460 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((59489,41406,19944)\) → \((e\left(\frac{13}{14}\right),e\left(\frac{3}{5}\right),e\left(\frac{11}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(15\)
\( \chi_{ 91091 }(20, a) \) \(1\)\(1\)\(e\left(\frac{4441}{5460}\right)\)\(e\left(\frac{1289}{2730}\right)\)\(e\left(\frac{1711}{2730}\right)\)\(e\left(\frac{1753}{1820}\right)\)\(e\left(\frac{1559}{5460}\right)\)\(e\left(\frac{801}{1820}\right)\)\(e\left(\frac{1289}{1365}\right)\)\(e\left(\frac{212}{273}\right)\)\(e\left(\frac{9}{91}\right)\)\(e\left(\frac{2377}{5460}\right)\)
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)
 
\( \chi_{ 91091 }(20,a) \;\) at \(\;a = \) e.g. 2