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Properties

Label 12397.578
Modulus $12397$
Conductor $12397$
Order $2310$
Real no
Primitive yes
Minimal yes
Parity odd

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(12397, base_ring=CyclotomicField(2310)) M = H._module chi = DirichletCharacter(H, M([1870,2079,1680]))
 
Copy content gp:[g,chi] = znchar(Mod(578, 12397))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12397.578");
 

Basic properties

Modulus: \(12397\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12397\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2310\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 12397.eu

\(\chi_{12397}(2,\cdot)\) \(\chi_{12397}(39,\cdot)\) \(\chi_{12397}(72,\cdot)\) \(\chi_{12397}(95,\cdot)\) \(\chi_{12397}(123,\cdot)\) \(\chi_{12397}(151,\cdot)\) \(\chi_{12397}(156,\cdot)\) \(\chi_{12397}(193,\cdot)\) \(\chi_{12397}(200,\cdot)\) \(\chi_{12397}(233,\cdot)\) \(\chi_{12397}(261,\cdot)\) \(\chi_{12397}(282,\cdot)\) \(\chi_{12397}(303,\cdot)\) \(\chi_{12397}(305,\cdot)\) \(\chi_{12397}(326,\cdot)\) \(\chi_{12397}(338,\cdot)\) \(\chi_{12397}(347,\cdot)\) \(\chi_{12397}(354,\cdot)\) \(\chi_{12397}(380,\cdot)\) \(\chi_{12397}(403,\cdot)\) \(\chi_{12397}(464,\cdot)\) \(\chi_{12397}(492,\cdot)\) \(\chi_{12397}(501,\cdot)\) \(\chi_{12397}(541,\cdot)\) \(\chi_{12397}(578,\cdot)\) \(\chi_{12397}(611,\cdot)\) \(\chi_{12397}(634,\cdot)\) \(\chi_{12397}(646,\cdot)\) \(\chi_{12397}(662,\cdot)\) \(\chi_{12397}(739,\cdot)\) ...

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: $\Q(\zeta_{1155})$
Fixed field: Number field defined by a degree 2310 polynomial (not computed)

Values on generators

\((9362,10144,2696)\) → \((e\left(\frac{17}{21}\right),e\left(\frac{9}{10}\right),e\left(\frac{8}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 12397 }(578, a) \) \(-1\)\(1\)\(e\left(\frac{929}{2310}\right)\)\(e\left(\frac{746}{1155}\right)\)\(e\left(\frac{929}{1155}\right)\)\(e\left(\frac{928}{1155}\right)\)\(e\left(\frac{37}{770}\right)\)\(e\left(\frac{159}{770}\right)\)\(e\left(\frac{337}{1155}\right)\)\(e\left(\frac{95}{462}\right)\)\(e\left(\frac{104}{231}\right)\)\(e\left(\frac{613}{770}\right)\)
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)
 
\( \chi_{ 12397 }(578,a) \;\) at \(\;a = \) e.g. 2