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Properties

Label 85600.219
Modulus $85600$
Conductor $85600$
Order $2120$
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(85600, base_ring=CyclotomicField(2120)) M = H._module chi = DirichletCharacter(H, M([1060,265,1908,940]))
 
Copy content gp:[g,chi] = znchar(Mod(219, 85600))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85600.219");
 

Basic properties

Modulus: \(85600\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(85600\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2120\)
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 85600.ig

\(\chi_{85600}(59,\cdot)\) \(\chi_{85600}(139,\cdot)\) \(\chi_{85600}(179,\cdot)\) \(\chi_{85600}(219,\cdot)\) \(\chi_{85600}(259,\cdot)\) \(\chi_{85600}(339,\cdot)\) \(\chi_{85600}(379,\cdot)\) \(\chi_{85600}(419,\cdot)\) \(\chi_{85600}(459,\cdot)\) \(\chi_{85600}(619,\cdot)\) \(\chi_{85600}(659,\cdot)\) \(\chi_{85600}(739,\cdot)\) \(\chi_{85600}(819,\cdot)\) \(\chi_{85600}(1059,\cdot)\) \(\chi_{85600}(1179,\cdot)\) \(\chi_{85600}(1259,\cdot)\) \(\chi_{85600}(1339,\cdot)\) \(\chi_{85600}(1379,\cdot)\) \(\chi_{85600}(1419,\cdot)\) \(\chi_{85600}(1459,\cdot)\)

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: $\Q(\zeta_{2120})$
Fixed field: Number field defined by a degree 2120 polynomial (not computed)

Values on generators

\((26751,32101,82177,16801)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{9}{10}\right),e\left(\frac{47}{106}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 85600 }(219, a) \) \(1\)\(1\)\(e\left(\frac{451}{2120}\right)\)\(e\left(\frac{67}{212}\right)\)\(e\left(\frac{451}{1060}\right)\)\(e\left(\frac{593}{2120}\right)\)\(e\left(\frac{387}{2120}\right)\)\(e\left(\frac{31}{530}\right)\)\(e\left(\frac{339}{2120}\right)\)\(e\left(\frac{1121}{2120}\right)\)\(e\left(\frac{679}{1060}\right)\)\(e\left(\frac{1353}{2120}\right)\)
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)
 
\( \chi_{ 85600 }(219,a) \;\) at \(\;a = \) e.g. 2