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Properties

Label 1581.110
Modulus $1581$
Conductor $1581$
Order $120$
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(1581, base_ring=CyclotomicField(120)) M = H._module chi = DirichletCharacter(H, M([60,75,28]))
 
Copy content gp:[g,chi] = znchar(Mod(110, 1581))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1581.110");
 

Basic properties

Modulus: \(1581\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1581\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(120\)
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 1581.cx

\(\chi_{1581}(53,\cdot)\) \(\chi_{1581}(83,\cdot)\) \(\chi_{1581}(104,\cdot)\) \(\chi_{1581}(110,\cdot)\) \(\chi_{1581}(179,\cdot)\) \(\chi_{1581}(230,\cdot)\) \(\chi_{1581}(332,\cdot)\) \(\chi_{1581}(365,\cdot)\) \(\chi_{1581}(383,\cdot)\) \(\chi_{1581}(389,\cdot)\) \(\chi_{1581}(416,\cdot)\) \(\chi_{1581}(518,\cdot)\) \(\chi_{1581}(569,\cdot)\) \(\chi_{1581}(644,\cdot)\) \(\chi_{1581}(695,\cdot)\) \(\chi_{1581}(716,\cdot)\) \(\chi_{1581}(797,\cdot)\) \(\chi_{1581}(818,\cdot)\) \(\chi_{1581}(848,\cdot)\) \(\chi_{1581}(920,\cdot)\) \(\chi_{1581}(995,\cdot)\) \(\chi_{1581}(1097,\cdot)\) \(\chi_{1581}(1181,\cdot)\) \(\chi_{1581}(1199,\cdot)\) \(\chi_{1581}(1226,\cdot)\) \(\chi_{1581}(1283,\cdot)\) \(\chi_{1581}(1385,\cdot)\) \(\chi_{1581}(1460,\cdot)\) \(\chi_{1581}(1481,\cdot)\) \(\chi_{1581}(1505,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((1055,1210,1429)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{7}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1581 }(110, a) \) \(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{49}{120}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{17}{120}\right)\)\(e\left(\frac{29}{120}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{31}{120}\right)\)\(e\left(\frac{2}{5}\right)\)
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)
 
\( \chi_{ 1581 }(110,a) \;\) at \(\;a = \) e.g. 2