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Properties

Label 4179.1244
Modulus $4179$
Conductor $4179$
Order $198$
Real no
Primitive yes
Minimal yes
Parity even

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(4179, base_ring=CyclotomicField(198)) M = H._module chi = DirichletCharacter(H, M([99,165,184]))
 
Copy content gp:[g,chi] = znchar(Mod(1244, 4179))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4179.1244");
 

Basic properties

Modulus: \(4179\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4179\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(198\)
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 4179.ed

\(\chi_{4179}(47,\cdot)\) \(\chi_{4179}(89,\cdot)\) \(\chi_{4179}(215,\cdot)\) \(\chi_{4179}(248,\cdot)\) \(\chi_{4179}(269,\cdot)\) \(\chi_{4179}(278,\cdot)\) \(\chi_{4179}(290,\cdot)\) \(\chi_{4179}(311,\cdot)\) \(\chi_{4179}(383,\cdot)\) \(\chi_{4179}(500,\cdot)\) \(\chi_{4179}(677,\cdot)\) \(\chi_{4179}(719,\cdot)\) \(\chi_{4179}(752,\cdot)\) \(\chi_{4179}(803,\cdot)\) \(\chi_{4179}(992,\cdot)\) \(\chi_{4179}(1004,\cdot)\) \(\chi_{4179}(1067,\cdot)\) \(\chi_{4179}(1076,\cdot)\) \(\chi_{4179}(1160,\cdot)\) \(\chi_{4179}(1214,\cdot)\) \(\chi_{4179}(1223,\cdot)\) \(\chi_{4179}(1244,\cdot)\) \(\chi_{4179}(1424,\cdot)\) \(\chi_{4179}(1487,\cdot)\) \(\chi_{4179}(1517,\cdot)\) \(\chi_{4179}(1538,\cdot)\) \(\chi_{4179}(1643,\cdot)\) \(\chi_{4179}(1769,\cdot)\) \(\chi_{4179}(1823,\cdot)\) \(\chi_{4179}(1844,\cdot)\) ...

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: $\Q(\zeta_{99})$
Fixed field: Number field defined by a degree 198 polynomial (not computed)

Values on generators

\((1394,598,799)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{92}{99}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4179 }(1244, a) \) \(1\)\(1\)\(e\left(\frac{133}{198}\right)\)\(e\left(\frac{34}{99}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{1}{66}\right)\)\(e\left(\frac{115}{198}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{67}{198}\right)\)\(e\left(\frac{68}{99}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{18}\right)\)
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)
 
\( \chi_{ 4179 }(1244,a) \;\) at \(\;a = \) e.g. 2