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Properties

Label 195083.1259
Modulus $195083$
Conductor $195083$
Order $1860$
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(195083, base_ring=CyclotomicField(1860)) M = H._module chi = DirichletCharacter(H, M([930,465,1328]))
 
Copy content gp:[g,chi] = znchar(Mod(1259, 195083))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("195083.1259");
 

Basic properties

Modulus: \(195083\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(195083\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1860\)
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 195083.pc

\(\chi_{195083}(41,\cdot)\) \(\chi_{195083}(307,\cdot)\) \(\chi_{195083}(510,\cdot)\) \(\chi_{195083}(1259,\cdot)\) \(\chi_{195083}(1322,\cdot)\) \(\chi_{195083}(1931,\cdot)\) \(\chi_{195083}(2477,\cdot)\) \(\chi_{195083}(2680,\cdot)\) \(\chi_{195083}(3149,\cdot)\) \(\chi_{195083}(3492,\cdot)\) \(\chi_{195083}(3758,\cdot)\) \(\chi_{195083}(4101,\cdot)\) \(\chi_{195083}(4164,\cdot)\) \(\chi_{195083}(5319,\cdot)\) \(\chi_{195083}(5382,\cdot)\) \(\chi_{195083}(5928,\cdot)\) \(\chi_{195083}(6334,\cdot)\) \(\chi_{195083}(6600,\cdot)\) \(\chi_{195083}(6803,\cdot)\) \(\chi_{195083}(7552,\cdot)\) \(\chi_{195083}(7615,\cdot)\) \(\chi_{195083}(8224,\cdot)\) \(\chi_{195083}(8770,\cdot)\) \(\chi_{195083}(8973,\cdot)\) \(\chi_{195083}(9442,\cdot)\) \(\chi_{195083}(9785,\cdot)\) \(\chi_{195083}(10051,\cdot)\) \(\chi_{195083}(10394,\cdot)\) \(\chi_{195083}(10457,\cdot)\) \(\chi_{195083}(11612,\cdot)\) ...

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: $\Q(\zeta_{1860})$
Fixed field: Number field defined by a degree 1860 polynomial (not computed)

Values on generators

\((27870,188357,180671)\) → \((-1,i,e\left(\frac{332}{465}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 195083 }(1259, a) \) \(1\)\(1\)\(e\left(\frac{479}{620}\right)\)\(e\left(\frac{863}{1860}\right)\)\(e\left(\frac{169}{310}\right)\)\(e\left(\frac{14}{93}\right)\)\(e\left(\frac{22}{93}\right)\)\(e\left(\frac{197}{620}\right)\)\(e\left(\frac{863}{930}\right)\)\(e\left(\frac{1717}{1860}\right)\)\(e\left(\frac{1609}{1860}\right)\)\(e\left(\frac{17}{1860}\right)\)
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)
 
\( \chi_{ 195083 }(1259,a) \;\) at \(\;a = \) e.g. 2