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Properties

Label 1849.178
Modulus $1849$
Conductor $1849$
Order $129$
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(1849, base_ring=CyclotomicField(258)) M = H._module chi = DirichletCharacter(H, M([136]))
 
Copy content gp:[g,chi] = znchar(Mod(178, 1849))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1849.178");
 

Basic properties

Modulus: \(1849\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1849\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(129\)
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 1849.k

\(\chi_{1849}(6,\cdot)\) \(\chi_{1849}(36,\cdot)\) \(\chi_{1849}(49,\cdot)\) \(\chi_{1849}(79,\cdot)\) \(\chi_{1849}(92,\cdot)\) \(\chi_{1849}(122,\cdot)\) \(\chi_{1849}(135,\cdot)\) \(\chi_{1849}(165,\cdot)\) \(\chi_{1849}(178,\cdot)\) \(\chi_{1849}(208,\cdot)\) \(\chi_{1849}(221,\cdot)\) \(\chi_{1849}(251,\cdot)\) \(\chi_{1849}(264,\cdot)\) \(\chi_{1849}(294,\cdot)\) \(\chi_{1849}(307,\cdot)\) \(\chi_{1849}(337,\cdot)\) \(\chi_{1849}(350,\cdot)\) \(\chi_{1849}(380,\cdot)\) \(\chi_{1849}(393,\cdot)\) \(\chi_{1849}(436,\cdot)\) \(\chi_{1849}(466,\cdot)\) \(\chi_{1849}(479,\cdot)\) \(\chi_{1849}(509,\cdot)\) \(\chi_{1849}(522,\cdot)\) \(\chi_{1849}(552,\cdot)\) \(\chi_{1849}(565,\cdot)\) \(\chi_{1849}(595,\cdot)\) \(\chi_{1849}(608,\cdot)\) \(\chi_{1849}(638,\cdot)\) \(\chi_{1849}(651,\cdot)\) ...

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: $\Q(\zeta_{129})$
Fixed field: Number field defined by a degree 129 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{68}{129}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1849 }(178, a) \) \(1\)\(1\)\(e\left(\frac{11}{43}\right)\)\(e\left(\frac{68}{129}\right)\)\(e\left(\frac{22}{43}\right)\)\(e\left(\frac{95}{129}\right)\)\(e\left(\frac{101}{129}\right)\)\(e\left(\frac{19}{129}\right)\)\(e\left(\frac{33}{43}\right)\)\(e\left(\frac{7}{129}\right)\)\(e\left(\frac{128}{129}\right)\)\(e\left(\frac{24}{43}\right)\)
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)
 
\( \chi_{ 1849 }(178,a) \;\) at \(\;a = \) e.g. 2