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Properties

Label 1919.499
Modulus $1919$
Conductor $1919$
Order $45$
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(1919, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([80,18]))
 
Copy content gp:[g,chi] = znchar(Mod(499, 1919))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1919.499");
 

Basic properties

Modulus: \(1919\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1919\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(45\)
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 1919.bd

\(\chi_{1919}(36,\cdot)\) \(\chi_{1919}(137,\cdot)\) \(\chi_{1919}(188,\cdot)\) \(\chi_{1919}(196,\cdot)\) \(\chi_{1919}(289,\cdot)\) \(\chi_{1919}(339,\cdot)\) \(\chi_{1919}(491,\cdot)\) \(\chi_{1919}(499,\cdot)\) \(\chi_{1919}(541,\cdot)\) \(\chi_{1919}(690,\cdot)\) \(\chi_{1919}(693,\cdot)\) \(\chi_{1919}(701,\cdot)\) \(\chi_{1919}(802,\cdot)\) \(\chi_{1919}(993,\cdot)\) \(\chi_{1919}(1004,\cdot)\) \(\chi_{1919}(1195,\cdot)\) \(\chi_{1919}(1206,\cdot)\) \(\chi_{1919}(1296,\cdot)\) \(\chi_{1919}(1450,\cdot)\) \(\chi_{1919}(1498,\cdot)\) \(\chi_{1919}(1602,\cdot)\) \(\chi_{1919}(1700,\cdot)\) \(\chi_{1919}(1753,\cdot)\) \(\chi_{1919}(1905,\cdot)\)

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 45 polynomial

Values on generators

\((1617,305)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1919 }(499, a) \) \(1\)\(1\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{4}{15}\right)\)
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)
 
\( \chi_{ 1919 }(499,a) \;\) at \(\;a = \) e.g. 2