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Properties

Label 181447.29
Modulus $181447$
Conductor $181447$
Order $12397$
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(181447, base_ring=CyclotomicField(24794)) M = H._module chi = DirichletCharacter(H, M([19228,7350]))
 
Copy content gp:[g,chi] = znchar(Mod(29, 181447))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("181447.29");
 

Basic properties

Modulus: \(181447\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(181447\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(12397\)
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 181447.dk

\(\chi_{181447}(8,\cdot)\) \(\chi_{181447}(29,\cdot)\) \(\chi_{181447}(36,\cdot)\) \(\chi_{181447}(64,\cdot)\) \(\chi_{181447}(71,\cdot)\) \(\chi_{181447}(78,\cdot)\) \(\chi_{181447}(85,\cdot)\) \(\chi_{181447}(127,\cdot)\) \(\chi_{181447}(141,\cdot)\) \(\chi_{181447}(169,\cdot)\) \(\chi_{181447}(190,\cdot)\) \(\chi_{181447}(211,\cdot)\) \(\chi_{181447}(225,\cdot)\) \(\chi_{181447}(232,\cdot)\) \(\chi_{181447}(239,\cdot)\) \(\chi_{181447}(288,\cdot)\) \(\chi_{181447}(302,\cdot)\) \(\chi_{181447}(330,\cdot)\) \(\chi_{181447}(351,\cdot)\) \(\chi_{181447}(358,\cdot)\) \(\chi_{181447}(372,\cdot)\) \(\chi_{181447}(386,\cdot)\) \(\chi_{181447}(400,\cdot)\) \(\chi_{181447}(407,\cdot)\) \(\chi_{181447}(449,\cdot)\) \(\chi_{181447}(463,\cdot)\) \(\chi_{181447}(512,\cdot)\) \(\chi_{181447}(519,\cdot)\) \(\chi_{181447}(533,\cdot)\) \(\chi_{181447}(547,\cdot)\) ...

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: $\Q(\zeta_{12397})$
Fixed field: Number field defined by a degree 12397 polynomial (not computed)

Values on generators

\((150237,62427)\) → \((e\left(\frac{38}{49}\right),e\left(\frac{75}{253}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 181447 }(29, a) \) \(1\)\(1\)\(e\left(\frac{9143}{12397}\right)\)\(e\left(\frac{6429}{12397}\right)\)\(e\left(\frac{5889}{12397}\right)\)\(e\left(\frac{9747}{12397}\right)\)\(e\left(\frac{3175}{12397}\right)\)\(e\left(\frac{2635}{12397}\right)\)\(e\left(\frac{461}{12397}\right)\)\(e\left(\frac{6493}{12397}\right)\)\(e\left(\frac{5069}{12397}\right)\)\(e\left(\frac{12318}{12397}\right)\)
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)
 
\( \chi_{ 181447 }(29,a) \;\) at \(\;a = \) e.g. 2