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Properties

Label 72111.659
Modulus $72111$
Conductor $72111$
Order $1806$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72111, base_ring=CyclotomicField(1806)) M = H._module chi = DirichletCharacter(H, M([903,1204,1742]))
 
Copy content gp:[g,chi] = znchar(Mod(659, 72111))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("72111.659");
 

Basic properties

Modulus: \(72111\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(72111\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1806\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 72111.ig

\(\chi_{72111}(230,\cdot)\) \(\chi_{72111}(341,\cdot)\) \(\chi_{72111}(497,\cdot)\) \(\chi_{72111}(659,\cdot)\) \(\chi_{72111}(848,\cdot)\) \(\chi_{72111}(926,\cdot)\) \(\chi_{72111}(971,\cdot)\) \(\chi_{72111}(1049,\cdot)\) \(\chi_{72111}(1088,\cdot)\) \(\chi_{72111}(1127,\cdot)\) \(\chi_{72111}(1199,\cdot)\) \(\chi_{72111}(1472,\cdot)\)

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: $\Q(\zeta_{903})$
Fixed field: Number field defined by a degree 1806 polynomial (not computed)

Values on generators

\((24038,16642,62869)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{871}{903}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 72111 }(659, a) \) \(-1\)\(1\)\(e\left(\frac{1135}{1806}\right)\)\(e\left(\frac{232}{903}\right)\)\(e\left(\frac{1193}{1806}\right)\)\(e\left(\frac{28}{43}\right)\)\(e\left(\frac{533}{602}\right)\)\(e\left(\frac{87}{301}\right)\)\(e\left(\frac{901}{1806}\right)\)\(e\left(\frac{505}{1806}\right)\)\(e\left(\frac{464}{903}\right)\)\(e\left(\frac{97}{602}\right)\)
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)
 
\( \chi_{ 72111 }(659,a) \;\) at \(\;a = \) e.g. 2