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Properties

Label 23275.1117
Modulus $23275$
Conductor $23275$
Order $1260$
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(23275, base_ring=CyclotomicField(1260)) M = H._module chi = DirichletCharacter(H, M([819,1020,770]))
 
Copy content gp:[g,chi] = znchar(Mod(1117, 23275))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("23275.1117");
 

Basic properties

Modulus: \(23275\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(23275\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1260\)
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 23275.oq

\(\chi_{23275}(2,\cdot)\) \(\chi_{23275}(53,\cdot)\) \(\chi_{23275}(212,\cdot)\) \(\chi_{23275}(298,\cdot)\) \(\chi_{23275}(333,\cdot)\) \(\chi_{23275}(452,\cdot)\) \(\chi_{23275}(478,\cdot)\) \(\chi_{23275}(527,\cdot)\) \(\chi_{23275}(697,\cdot)\) \(\chi_{23275}(877,\cdot)\) \(\chi_{23275}(933,\cdot)\) \(\chi_{23275}(963,\cdot)\) \(\chi_{23275}(1117,\cdot)\) \(\chi_{23275}(1192,\cdot)\) \(\chi_{23275}(1362,\cdot)\) \(\chi_{23275}(1383,\cdot)\) \(\chi_{23275}(1397,\cdot)\) \(\chi_{23275}(1458,\cdot)\) \(\chi_{23275}(1542,\cdot)\)

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: $\Q(\zeta_{1260})$
Fixed field: Number field defined by a degree 1260 polynomial (not computed)

Values on generators

\((19552,12351,20826)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{17}{21}\right),e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 23275 }(1117, a) \) \(1\)\(1\)\(e\left(\frac{389}{1260}\right)\)\(e\left(\frac{383}{1260}\right)\)\(e\left(\frac{389}{630}\right)\)\(e\left(\frac{193}{315}\right)\)\(e\left(\frac{389}{420}\right)\)\(e\left(\frac{383}{630}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{129}{140}\right)\)\(e\left(\frac{151}{1260}\right)\)\(e\left(\frac{74}{315}\right)\)
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)
 
\( \chi_{ 23275 }(1117,a) \;\) at \(\;a = \) e.g. 2