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Properties

Label 4444.887
Modulus $4444$
Conductor $4444$
Order $50$
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(4444, base_ring=CyclotomicField(50)) M = H._module chi = DirichletCharacter(H, M([25,35,32]))
 
Copy content gp:[g,chi] = znchar(Mod(887, 4444))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4444.887");
 

Basic properties

Modulus: \(4444\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4444\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(50\)
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 4444.dm

\(\chi_{4444}(19,\cdot)\) \(\chi_{4444}(239,\cdot)\) \(\chi_{4444}(283,\cdot)\) \(\chi_{4444}(299,\cdot)\) \(\chi_{4444}(563,\cdot)\) \(\chi_{4444}(787,\cdot)\) \(\chi_{4444}(887,\cdot)\) \(\chi_{4444}(963,\cdot)\) \(\chi_{4444}(987,\cdot)\) \(\chi_{4444}(1135,\cdot)\) \(\chi_{4444}(1179,\cdot)\) \(\chi_{4444}(1283,\cdot)\) \(\chi_{4444}(1603,\cdot)\) \(\chi_{4444}(1843,\cdot)\) \(\chi_{4444}(2339,\cdot)\) \(\chi_{4444}(2415,\cdot)\) \(\chi_{4444}(3263,\cdot)\) \(\chi_{4444}(3439,\cdot)\) \(\chi_{4444}(3995,\cdot)\) \(\chi_{4444}(4395,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((2223,2829,2729)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{16}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 4444 }(887, a) \) \(1\)\(1\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{21}{50}\right)\)\(-1\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{21}{50}\right)\)\(e\left(\frac{27}{50}\right)\)
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)
 
\( \chi_{ 4444 }(887,a) \;\) at \(\;a = \) e.g. 2