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Properties

Label 6105.47
Modulus $6105$
Conductor $6105$
Order $60$
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(6105, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([30,15,48,40]))
 
Copy content gp:[g,chi] = znchar(Mod(47, 6105))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6105.47");
 

Basic properties

Modulus: \(6105\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6105\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(60\)
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 6105.he

\(\chi_{6105}(47,\cdot)\) \(\chi_{6105}(137,\cdot)\) \(\chi_{6105}(158,\cdot)\) \(\chi_{6105}(713,\cdot)\) \(\chi_{6105}(1247,\cdot)\) \(\chi_{6105}(1268,\cdot)\) \(\chi_{6105}(1358,\cdot)\) \(\chi_{6105}(1802,\cdot)\) \(\chi_{6105}(2357,\cdot)\) \(\chi_{6105}(2468,\cdot)\) \(\chi_{6105}(3023,\cdot)\) \(\chi_{6105}(3578,\cdot)\) \(\chi_{6105}(3932,\cdot)\) \(\chi_{6105}(5042,\cdot)\) \(\chi_{6105}(5153,\cdot)\) \(\chi_{6105}(5597,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((2036,1222,4996,3961)\) → \((-1,i,e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\( \chi_{ 6105 }(47, a) \) \(1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(i\)
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)
 
\( \chi_{ 6105 }(47,a) \;\) at \(\;a = \) e.g. 2