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Properties

Label 52675.14193
Modulus $52675$
Conductor $10535$
Order $84$
Real no
Primitive no
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(52675, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([63,8,2]))
 
Copy content gp:[g,chi] = znchar(Mod(14193, 52675))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("52675.14193");
 

Basic properties

Modulus: \(52675\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(10535\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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: no, induced from \(\chi_{10535}(3658,\cdot)\)
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 52675.wr

\(\chi_{52675}(893,\cdot)\) \(\chi_{52675}(2643,\cdot)\) \(\chi_{52675}(5107,\cdot)\) \(\chi_{52675}(6393,\cdot)\) \(\chi_{52675}(6857,\cdot)\) \(\chi_{52675}(10607,\cdot)\) \(\chi_{52675}(10768,\cdot)\) \(\chi_{52675}(14193,\cdot)\) \(\chi_{52675}(14982,\cdot)\) \(\chi_{52675}(18407,\cdot)\) \(\chi_{52675}(21268,\cdot)\) \(\chi_{52675}(25482,\cdot)\) \(\chi_{52675}(26518,\cdot)\) \(\chi_{52675}(30732,\cdot)\) \(\chi_{52675}(37293,\cdot)\) \(\chi_{52675}(39293,\cdot)\) \(\chi_{52675}(41507,\cdot)\) \(\chi_{52675}(42718,\cdot)\) \(\chi_{52675}(43507,\cdot)\) \(\chi_{52675}(46468,\cdot)\) \(\chi_{52675}(46918,\cdot)\) \(\chi_{52675}(46932,\cdot)\) \(\chi_{52675}(50682,\cdot)\) \(\chi_{52675}(51132,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((37927,34401,47776)\) → \((-i,e\left(\frac{2}{21}\right),e\left(\frac{1}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 52675 }(14193, a) \) \(1\)\(1\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{10}{21}\right)\)
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)
 
\( \chi_{ 52675 }(14193,a) \;\) at \(\;a = \) e.g. 2