↑

Properties

Label 4027.89
Modulus $4027$
Conductor $4027$
Order $2013$
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(4027, base_ring=CyclotomicField(4026)) M = H._module chi = DirichletCharacter(H, M([2188]))
 
Copy content gp:[g,chi] = znchar(Mod(89, 4027))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4027.89");
 

Basic properties

Modulus: \(4027\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4027\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2013\)
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 4027.o

\(\chi_{4027}(6,\cdot)\) \(\chi_{4027}(9,\cdot)\) \(\chi_{4027}(10,\cdot)\) \(\chi_{4027}(17,\cdot)\) \(\chi_{4027}(19,\cdot)\) \(\chi_{4027}(21,\cdot)\) \(\chi_{4027}(22,\cdot)\) \(\chi_{4027}(24,\cdot)\) \(\chi_{4027}(25,\cdot)\) \(\chi_{4027}(29,\cdot)\) \(\chi_{4027}(35,\cdot)\) \(\chi_{4027}(36,\cdot)\) \(\chi_{4027}(40,\cdot)\) \(\chi_{4027}(46,\cdot)\) \(\chi_{4027}(53,\cdot)\) \(\chi_{4027}(55,\cdot)\) \(\chi_{4027}(62,\cdot)\) \(\chi_{4027}(71,\cdot)\) \(\chi_{4027}(74,\cdot)\) \(\chi_{4027}(76,\cdot)\) \(\chi_{4027}(77,\cdot)\) \(\chi_{4027}(78,\cdot)\) \(\chi_{4027}(79,\cdot)\) \(\chi_{4027}(81,\cdot)\) \(\chi_{4027}(83,\cdot)\) \(\chi_{4027}(84,\cdot)\) \(\chi_{4027}(88,\cdot)\) \(\chi_{4027}(89,\cdot)\) \(\chi_{4027}(90,\cdot)\) \(\chi_{4027}(96,\cdot)\) ...

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: $\Q(\zeta_{2013})$
Fixed field: Number field defined by a degree 2013 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1094}{2013}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4027 }(89, a) \) \(1\)\(1\)\(e\left(\frac{537}{671}\right)\)\(e\left(\frac{1094}{2013}\right)\)\(e\left(\frac{403}{671}\right)\)\(e\left(\frac{1093}{2013}\right)\)\(e\left(\frac{692}{2013}\right)\)\(e\left(\frac{439}{671}\right)\)\(e\left(\frac{269}{671}\right)\)\(e\left(\frac{175}{2013}\right)\)\(e\left(\frac{691}{2013}\right)\)\(e\left(\frac{1885}{2013}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 4027 }(89,a) \;\) at \(\;a = \) e.g. 2