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Properties

Label 3755.1543
Modulus $3755$
Conductor $3755$
Order $100$
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(3755, base_ring=CyclotomicField(100)) M = H._module chi = DirichletCharacter(H, M([75,62]))
 
Copy content gp:[g,chi] = znchar(Mod(1543, 3755))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3755.1543");
 

Basic properties

Modulus: \(3755\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3755\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(100\)
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 3755.be

\(\chi_{3755}(48,\cdot)\) \(\chi_{3755}(252,\cdot)\) \(\chi_{3755}(403,\cdot)\) \(\chi_{3755}(558,\cdot)\) \(\chi_{3755}(572,\cdot)\) \(\chi_{3755}(698,\cdot)\) \(\chi_{3755}(792,\cdot)\) \(\chi_{3755}(1003,\cdot)\) \(\chi_{3755}(1017,\cdot)\) \(\chi_{3755}(1027,\cdot)\) \(\chi_{3755}(1052,\cdot)\) \(\chi_{3755}(1082,\cdot)\) \(\chi_{3755}(1177,\cdot)\) \(\chi_{3755}(1323,\cdot)\) \(\chi_{3755}(1543,\cdot)\) \(\chi_{3755}(1587,\cdot)\) \(\chi_{3755}(1697,\cdot)\) \(\chi_{3755}(1768,\cdot)\) \(\chi_{3755}(1772,\cdot)\) \(\chi_{3755}(1778,\cdot)\) \(\chi_{3755}(1787,\cdot)\) \(\chi_{3755}(1803,\cdot)\) \(\chi_{3755}(1833,\cdot)\) \(\chi_{3755}(1928,\cdot)\) \(\chi_{3755}(2082,\cdot)\) \(\chi_{3755}(2202,\cdot)\) \(\chi_{3755}(2338,\cdot)\) \(\chi_{3755}(2448,\cdot)\) \(\chi_{3755}(2523,\cdot)\) \(\chi_{3755}(2538,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((752,2256)\) → \((-i,e\left(\frac{31}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 3755 }(1543, a) \) \(1\)\(1\)\(e\left(\frac{67}{100}\right)\)\(e\left(\frac{87}{100}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{27}{50}\right)\)\(e\left(\frac{37}{100}\right)\)\(e\left(\frac{1}{100}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{21}{100}\right)\)\(e\left(\frac{21}{100}\right)\)
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)
 
\( \chi_{ 3755 }(1543,a) \;\) at \(\;a = \) e.g. 2