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Properties

Label 1199.1074
Modulus $1199$
Conductor $1199$
Order $90$
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(1199, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([63,55]))
 
Copy content gp:[g,chi] = znchar(Mod(1074, 1199))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1199.1074");
 

Basic properties

Modulus: \(1199\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1199\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
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 1199.bl

\(\chi_{1199}(222,\cdot)\) \(\chi_{1199}(261,\cdot)\) \(\chi_{1199}(370,\cdot)\) \(\chi_{1199}(398,\cdot)\) \(\chi_{1199}(409,\cdot)\) \(\chi_{1199}(420,\cdot)\) \(\chi_{1199}(470,\cdot)\) \(\chi_{1199}(479,\cdot)\) \(\chi_{1199}(579,\cdot)\) \(\chi_{1199}(688,\cdot)\) \(\chi_{1199}(767,\cdot)\) \(\chi_{1199}(876,\cdot)\) \(\chi_{1199}(915,\cdot)\) \(\chi_{1199}(943,\cdot)\) \(\chi_{1199}(954,\cdot)\) \(\chi_{1199}(965,\cdot)\) \(\chi_{1199}(985,\cdot)\) \(\chi_{1199}(1052,\cdot)\) \(\chi_{1199}(1063,\cdot)\) \(\chi_{1199}(1074,\cdot)\) \(\chi_{1199}(1124,\cdot)\) \(\chi_{1199}(1161,\cdot)\) \(\chi_{1199}(1172,\cdot)\) \(\chi_{1199}(1183,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((1091,551)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1199 }(1074, a) \) \(-1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{4}{9}\right)\)
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)
 
\( \chi_{ 1199 }(1074,a) \;\) at \(\;a = \) e.g. 2