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Properties

Label 2219.755
Modulus $2219$
Conductor $2219$
Order $158$
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(2219, base_ring=CyclotomicField(158)) M = H._module chi = DirichletCharacter(H, M([79,130]))
 
Copy content gp:[g,chi] = znchar(Mod(755, 2219))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2219.755");
 

Basic properties

Modulus: \(2219\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2219\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(158\)
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 2219.n

\(\chi_{2219}(34,\cdot)\) \(\chi_{2219}(104,\cdot)\) \(\chi_{2219}(160,\cdot)\) \(\chi_{2219}(181,\cdot)\) \(\chi_{2219}(223,\cdot)\) \(\chi_{2219}(230,\cdot)\) \(\chi_{2219}(244,\cdot)\) \(\chi_{2219}(251,\cdot)\) \(\chi_{2219}(328,\cdot)\) \(\chi_{2219}(384,\cdot)\) \(\chi_{2219}(398,\cdot)\) \(\chi_{2219}(482,\cdot)\) \(\chi_{2219}(496,\cdot)\) \(\chi_{2219}(510,\cdot)\) \(\chi_{2219}(538,\cdot)\) \(\chi_{2219}(552,\cdot)\) \(\chi_{2219}(573,\cdot)\) \(\chi_{2219}(594,\cdot)\) \(\chi_{2219}(608,\cdot)\) \(\chi_{2219}(650,\cdot)\) \(\chi_{2219}(657,\cdot)\) \(\chi_{2219}(685,\cdot)\) \(\chi_{2219}(699,\cdot)\) \(\chi_{2219}(713,\cdot)\) \(\chi_{2219}(734,\cdot)\) \(\chi_{2219}(755,\cdot)\) \(\chi_{2219}(783,\cdot)\) \(\chi_{2219}(790,\cdot)\) \(\chi_{2219}(839,\cdot)\) \(\chi_{2219}(846,\cdot)\) ...

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: $\Q(\zeta_{79})$
Fixed field: Number field defined by a degree 158 polynomial (not computed)

Values on generators

\((318,953)\) → \((-1,e\left(\frac{65}{79}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2219 }(755, a) \) \(-1\)\(1\)\(e\left(\frac{65}{79}\right)\)\(e\left(\frac{59}{158}\right)\)\(e\left(\frac{51}{79}\right)\)\(e\left(\frac{69}{158}\right)\)\(e\left(\frac{31}{158}\right)\)\(e\left(\frac{37}{79}\right)\)\(e\left(\frac{59}{79}\right)\)\(e\left(\frac{41}{158}\right)\)\(e\left(\frac{76}{79}\right)\)\(e\left(\frac{3}{158}\right)\)
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)
 
\( \chi_{ 2219 }(755,a) \;\) at \(\;a = \) e.g. 2