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Properties

Label 4553.69
Modulus $4553$
Conductor $4553$
Order $1092$
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(4553, base_ring=CyclotomicField(1092)) M = H._module chi = DirichletCharacter(H, M([975,427]))
 
Copy content gp:[g,chi] = znchar(Mod(69, 4553))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4553.69");
 

Basic properties

Modulus: \(4553\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4553\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1092\)
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 4553.cw

\(\chi_{4553}(15,\cdot)\) \(\chi_{4553}(21,\cdot)\) \(\chi_{4553}(26,\cdot)\) \(\chi_{4553}(43,\cdot)\) \(\chi_{4553}(60,\cdot)\) \(\chi_{4553}(61,\cdot)\) \(\chi_{4553}(66,\cdot)\) \(\chi_{4553}(69,\cdot)\) \(\chi_{4553}(84,\cdot)\) \(\chi_{4553}(97,\cdot)\) \(\chi_{4553}(102,\cdot)\) \(\chi_{4553}(114,\cdot)\) \(\chi_{4553}(137,\cdot)\) \(\chi_{4553}(163,\cdot)\) \(\chi_{4553}(172,\cdot)\) \(\chi_{4553}(195,\cdot)\) \(\chi_{4553}(217,\cdot)\) \(\chi_{4553}(229,\cdot)\) \(\chi_{4553}(240,\cdot)\) \(\chi_{4553}(251,\cdot)\) \(\chi_{4553}(259,\cdot)\) \(\chi_{4553}(271,\cdot)\) \(\chi_{4553}(276,\cdot)\) \(\chi_{4553}(329,\cdot)\) \(\chi_{4553}(334,\cdot)\) \(\chi_{4553}(338,\cdot)\) \(\chi_{4553}(369,\cdot)\) \(\chi_{4553}(388,\cdot)\) \(\chi_{4553}(408,\cdot)\) \(\chi_{4553}(409,\cdot)\) ...

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: $\Q(\zeta_{1092})$
Fixed field: Number field defined by a degree 1092 polynomial (not computed)

Values on generators

\((2670,2988)\) → \((e\left(\frac{25}{28}\right),e\left(\frac{61}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4553 }(69, a) \) \(1\)\(1\)\(e\left(\frac{5}{182}\right)\)\(e\left(\frac{577}{1092}\right)\)\(e\left(\frac{5}{91}\right)\)\(e\left(\frac{37}{1092}\right)\)\(e\left(\frac{607}{1092}\right)\)\(e\left(\frac{71}{364}\right)\)\(e\left(\frac{15}{182}\right)\)\(e\left(\frac{31}{546}\right)\)\(e\left(\frac{67}{1092}\right)\)\(e\left(\frac{295}{1092}\right)\)
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)
 
\( \chi_{ 4553 }(69,a) \;\) at \(\;a = \) e.g. 2