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Properties

Label 2873.1138
Modulus $2873$
Conductor $2873$
Order $156$
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(2873, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([23,78]))
 
Copy content gp:[g,chi] = znchar(Mod(1138, 2873))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2873.1138");
 

Basic properties

Modulus: \(2873\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2873\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(156\)
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 2873.cd

\(\chi_{2873}(33,\cdot)\) \(\chi_{2873}(50,\cdot)\) \(\chi_{2873}(67,\cdot)\) \(\chi_{2873}(84,\cdot)\) \(\chi_{2873}(254,\cdot)\) \(\chi_{2873}(271,\cdot)\) \(\chi_{2873}(288,\cdot)\) \(\chi_{2873}(305,\cdot)\) \(\chi_{2873}(475,\cdot)\) \(\chi_{2873}(492,\cdot)\) \(\chi_{2873}(509,\cdot)\) \(\chi_{2873}(696,\cdot)\) \(\chi_{2873}(713,\cdot)\) \(\chi_{2873}(730,\cdot)\) \(\chi_{2873}(747,\cdot)\) \(\chi_{2873}(917,\cdot)\) \(\chi_{2873}(951,\cdot)\) \(\chi_{2873}(968,\cdot)\) \(\chi_{2873}(1138,\cdot)\) \(\chi_{2873}(1155,\cdot)\) \(\chi_{2873}(1172,\cdot)\) \(\chi_{2873}(1189,\cdot)\) \(\chi_{2873}(1359,\cdot)\) \(\chi_{2873}(1376,\cdot)\) \(\chi_{2873}(1393,\cdot)\) \(\chi_{2873}(1410,\cdot)\) \(\chi_{2873}(1580,\cdot)\) \(\chi_{2873}(1597,\cdot)\) \(\chi_{2873}(1614,\cdot)\) \(\chi_{2873}(1631,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((171,2536)\) → \((e\left(\frac{23}{156}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2873 }(1138, a) \) \(-1\)\(1\)\(e\left(\frac{23}{156}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{23}{78}\right)\)\(e\left(\frac{43}{52}\right)\)\(e\left(\frac{145}{156}\right)\)\(e\left(\frac{43}{156}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{107}{156}\right)\)
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)
 
\( \chi_{ 2873 }(1138,a) \;\) at \(\;a = \) e.g. 2