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Properties

Label 379595.692
Modulus $379595$
Conductor $379595$
Order $24180$
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(379595, base_ring=CyclotomicField(24180)) M = H._module chi = DirichletCharacter(H, M([6045,17524,22010]))
 
Copy content gp:[g,chi] = znchar(Mod(692, 379595))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("379595.692");
 

Basic properties

Modulus: \(379595\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(379595\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(24180\)
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 379595.se

\(\chi_{379595}(28,\cdot)\) \(\chi_{379595}(107,\cdot)\) \(\chi_{379595}(138,\cdot)\) \(\chi_{379595}(193,\cdot)\) \(\chi_{379595}(267,\cdot)\) \(\chi_{379595}(297,\cdot)\) \(\chi_{379595}(443,\cdot)\) \(\chi_{379595}(472,\cdot)\) \(\chi_{379595}(638,\cdot)\) \(\chi_{379595}(692,\cdot)\) \(\chi_{379595}(702,\cdot)\) \(\chi_{379595}(908,\cdot)\) \(\chi_{379595}(917,\cdot)\) \(\chi_{379595}(1033,\cdot)\) \(\chi_{379595}(1228,\cdot)\) \(\chi_{379595}(1312,\cdot)\) \(\chi_{379595}(1378,\cdot)\) \(\chi_{379595}(1382,\cdot)\) \(\chi_{379595}(1402,\cdot)\) \(\chi_{379595}(1413,\cdot)\) \(\chi_{379595}(1497,\cdot)\) \(\chi_{379595}(1538,\cdot)\)

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: $\Q(\zeta_{24180})$
Fixed field: Number field defined by a degree 24180 polynomial (not computed)

Values on generators

\((303677,315211,216226)\) → \((i,e\left(\frac{337}{465}\right),e\left(\frac{71}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 379595 }(692, a) \) \(1\)\(1\)\(e\left(\frac{2981}{24180}\right)\)\(e\left(\frac{3103}{8060}\right)\)\(e\left(\frac{2981}{12090}\right)\)\(e\left(\frac{1229}{2418}\right)\)\(e\left(\frac{8867}{24180}\right)\)\(e\left(\frac{2981}{8060}\right)\)\(e\left(\frac{3103}{4030}\right)\)\(e\left(\frac{1986}{2015}\right)\)\(e\left(\frac{15271}{24180}\right)\)\(e\left(\frac{7639}{24180}\right)\)
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)
 
\( \chi_{ 379595 }(692,a) \;\) at \(\;a = \) e.g. 2