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Properties

Label 1151.44
Modulus $1151$
Conductor $1151$
Order $575$
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(1151, base_ring=CyclotomicField(1150)) M = H._module chi = DirichletCharacter(H, M([812]))
 
Copy content gp:[g,chi] = znchar(Mod(44, 1151))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1151.44");
 

Basic properties

Modulus: \(1151\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1151\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(575\)
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 1151.k

\(\chi_{1151}(2,\cdot)\) \(\chi_{1151}(3,\cdot)\) \(\chi_{1151}(4,\cdot)\) \(\chi_{1151}(5,\cdot)\) \(\chi_{1151}(8,\cdot)\) \(\chi_{1151}(9,\cdot)\) \(\chi_{1151}(10,\cdot)\) \(\chi_{1151}(11,\cdot)\) \(\chi_{1151}(12,\cdot)\) \(\chi_{1151}(14,\cdot)\) \(\chi_{1151}(16,\cdot)\) \(\chi_{1151}(18,\cdot)\) \(\chi_{1151}(20,\cdot)\) \(\chi_{1151}(21,\cdot)\) \(\chi_{1151}(22,\cdot)\) \(\chi_{1151}(24,\cdot)\) \(\chi_{1151}(25,\cdot)\) \(\chi_{1151}(27,\cdot)\) \(\chi_{1151}(28,\cdot)\) \(\chi_{1151}(29,\cdot)\) \(\chi_{1151}(30,\cdot)\) \(\chi_{1151}(33,\cdot)\) \(\chi_{1151}(35,\cdot)\) \(\chi_{1151}(37,\cdot)\) \(\chi_{1151}(40,\cdot)\) \(\chi_{1151}(44,\cdot)\) \(\chi_{1151}(45,\cdot)\) \(\chi_{1151}(48,\cdot)\) \(\chi_{1151}(50,\cdot)\) \(\chi_{1151}(53,\cdot)\) ...

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: $\Q(\zeta_{575})$
Fixed field: Number field defined by a degree 575 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{406}{575}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1151 }(44, a) \) \(1\)\(1\)\(e\left(\frac{118}{575}\right)\)\(e\left(\frac{162}{575}\right)\)\(e\left(\frac{236}{575}\right)\)\(e\left(\frac{473}{575}\right)\)\(e\left(\frac{56}{115}\right)\)\(e\left(\frac{43}{115}\right)\)\(e\left(\frac{354}{575}\right)\)\(e\left(\frac{324}{575}\right)\)\(e\left(\frac{16}{575}\right)\)\(e\left(\frac{536}{575}\right)\)
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)
 
\( \chi_{ 1151 }(44,a) \;\) at \(\;a = \) e.g. 2