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Properties

Label 106069.8707
Modulus $106069$
Conductor $106069$
Order $792$
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(106069, base_ring=CyclotomicField(792)) M = H._module chi = DirichletCharacter(H, M([187,372]))
 
Copy content gp:[g,chi] = znchar(Mod(8707, 106069))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("106069.8707");
 

Basic properties

Modulus: \(106069\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(106069\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(792\)
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 106069.jf

\(\chi_{106069}(756,\cdot)\) \(\chi_{106069}(1309,\cdot)\) \(\chi_{106069}(1572,\cdot)\) \(\chi_{106069}(1918,\cdot)\) \(\chi_{106069}(2785,\cdot)\) \(\chi_{106069}(2931,\cdot)\) \(\chi_{106069}(3038,\cdot)\) \(\chi_{106069}(3296,\cdot)\) \(\chi_{106069}(3327,\cdot)\) \(\chi_{106069}(3371,\cdot)\) \(\chi_{106069}(4248,\cdot)\) \(\chi_{106069}(4594,\cdot)\) \(\chi_{106069}(5115,\cdot)\) \(\chi_{106069}(5668,\cdot)\) \(\chi_{106069}(5801,\cdot)\) \(\chi_{106069}(6044,\cdot)\) \(\chi_{106069}(6758,\cdot)\) \(\chi_{106069}(7121,\cdot)\) \(\chi_{106069}(8574,\cdot)\) \(\chi_{106069}(8707,\cdot)\) \(\chi_{106069}(8953,\cdot)\) \(\chi_{106069}(8990,\cdot)\) \(\chi_{106069}(9139,\cdot)\) \(\chi_{106069}(10027,\cdot)\) \(\chi_{106069}(10060,\cdot)\) \(\chi_{106069}(10108,\cdot)\) \(\chi_{106069}(10160,\cdot)\) \(\chi_{106069}(10406,\cdot)\) \(\chi_{106069}(10540,\cdot)\) \(\chi_{106069}(11636,\cdot)\) ...

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: $\Q(\zeta_{792})$
Fixed field: Number field defined by a degree 792 polynomial (not computed)

Values on generators

\((90087,30515)\) → \((e\left(\frac{17}{72}\right),e\left(\frac{31}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 106069 }(8707, a) \) \(-1\)\(1\)\(e\left(\frac{71}{198}\right)\)\(e\left(\frac{107}{132}\right)\)\(e\left(\frac{71}{99}\right)\)\(e\left(\frac{5}{72}\right)\)\(e\left(\frac{67}{396}\right)\)\(e\left(\frac{89}{264}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{113}{264}\right)\)\(e\left(\frac{23}{72}\right)\)
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)
 
\( \chi_{ 106069 }(8707,a) \;\) at \(\;a = \) e.g. 2