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Properties

Label 1141.1006
Modulus $1141$
Conductor $1141$
Order $54$
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(1141, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([45,25]))
 
Copy content gp:[g,chi] = znchar(Mod(1006, 1141))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1141.1006");
 

Basic properties

Modulus: \(1141\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1141\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(54\)
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 1141.bp

\(\chi_{1141}(31,\cdot)\) \(\chi_{1141}(138,\cdot)\) \(\chi_{1141}(180,\cdot)\) \(\chi_{1141}(262,\cdot)\) \(\chi_{1141}(290,\cdot)\) \(\chi_{1141}(320,\cdot)\) \(\chi_{1141}(334,\cdot)\) \(\chi_{1141}(339,\cdot)\) \(\chi_{1141}(353,\cdot)\) \(\chi_{1141}(374,\cdot)\) \(\chi_{1141}(467,\cdot)\) \(\chi_{1141}(591,\cdot)\) \(\chi_{1141}(689,\cdot)\) \(\chi_{1141}(901,\cdot)\) \(\chi_{1141}(957,\cdot)\) \(\chi_{1141}(983,\cdot)\) \(\chi_{1141}(1006,\cdot)\) \(\chi_{1141}(1076,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((164,491)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{25}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1141 }(1006, a) \) \(1\)\(1\)\(e\left(\frac{7}{54}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{13}{54}\right)\)\(e\left(\frac{5}{54}\right)\)\(e\left(\frac{23}{27}\right)\)
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)
 
\( \chi_{ 1141 }(1006,a) \;\) at \(\;a = \) e.g. 2