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Properties

Label 11613.89
Modulus $11613$
Conductor $11613$
Order $546$
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(11613, base_ring=CyclotomicField(546)) M = H._module chi = DirichletCharacter(H, M([273,299,462]))
 
Copy content gp:[g,chi] = znchar(Mod(89, 11613))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11613.89");
 

Basic properties

Modulus: \(11613\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11613\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(546\)
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 11613.ft

\(\chi_{11613}(38,\cdot)\) \(\chi_{11613}(89,\cdot)\) \(\chi_{11613}(101,\cdot)\) \(\chi_{11613}(131,\cdot)\) \(\chi_{11613}(143,\cdot)\) \(\chi_{11613}(299,\cdot)\) \(\chi_{11613}(383,\cdot)\) \(\chi_{11613}(416,\cdot)\) \(\chi_{11613}(563,\cdot)\) \(\chi_{11613}(605,\cdot)\) \(\chi_{11613}(719,\cdot)\) \(\chi_{11613}(773,\cdot)\) \(\chi_{11613}(836,\cdot)\) \(\chi_{11613}(857,\cdot)\) \(\chi_{11613}(887,\cdot)\) \(\chi_{11613}(1013,\cdot)\) \(\chi_{11613}(1193,\cdot)\) \(\chi_{11613}(1223,\cdot)\) \(\chi_{11613}(1286,\cdot)\) \(\chi_{11613}(1328,\cdot)\) \(\chi_{11613}(1361,\cdot)\) \(\chi_{11613}(1487,\cdot)\) \(\chi_{11613}(1601,\cdot)\) \(\chi_{11613}(1748,\cdot)\) \(\chi_{11613}(1760,\cdot)\) \(\chi_{11613}(1790,\cdot)\) \(\chi_{11613}(1802,\cdot)\) \(\chi_{11613}(1958,\cdot)\) \(\chi_{11613}(2021,\cdot)\) \(\chi_{11613}(2042,\cdot)\) ...

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: $\Q(\zeta_{273})$
Fixed field: Number field defined by a degree 546 polynomial (not computed)

Values on generators

\((3872,2845,1030)\) → \((-1,e\left(\frac{23}{42}\right),e\left(\frac{11}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 11613 }(89, a) \) \(1\)\(1\)\(e\left(\frac{67}{546}\right)\)\(e\left(\frac{67}{273}\right)\)\(e\left(\frac{230}{273}\right)\)\(e\left(\frac{67}{182}\right)\)\(e\left(\frac{527}{546}\right)\)\(e\left(\frac{515}{546}\right)\)\(e\left(\frac{153}{182}\right)\)\(e\left(\frac{134}{273}\right)\)\(e\left(\frac{262}{273}\right)\)\(e\left(\frac{19}{78}\right)\)
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)
 
\( \chi_{ 11613 }(89,a) \;\) at \(\;a = \) e.g. 2