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Properties

Label 1991.1730
Modulus $1991$
Conductor $1991$
Order $90$
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(1991, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([72,35]))
 
Copy content gp:[g,chi] = znchar(Mod(1730, 1991))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1991.1730");
 

Basic properties

Modulus: \(1991\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1991\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1991.dx

\(\chi_{1991}(108,\cdot)\) \(\chi_{1991}(119,\cdot)\) \(\chi_{1991}(289,\cdot)\) \(\chi_{1991}(300,\cdot)\) \(\chi_{1991}(323,\cdot)\) \(\chi_{1991}(478,\cdot)\) \(\chi_{1991}(500,\cdot)\) \(\chi_{1991}(504,\cdot)\) \(\chi_{1991}(685,\cdot)\) \(\chi_{1991}(840,\cdot)\) \(\chi_{1991}(862,\cdot)\) \(\chi_{1991}(1006,\cdot)\) \(\chi_{1991}(1021,\cdot)\) \(\chi_{1991}(1043,\cdot)\) \(\chi_{1991}(1202,\cdot)\) \(\chi_{1991}(1224,\cdot)\) \(\chi_{1991}(1368,\cdot)\) \(\chi_{1991}(1549,\cdot)\) \(\chi_{1991}(1556,\cdot)\) \(\chi_{1991}(1567,\cdot)\) \(\chi_{1991}(1730,\cdot)\) \(\chi_{1991}(1918,\cdot)\) \(\chi_{1991}(1929,\cdot)\) \(\chi_{1991}(1952,\cdot)\)

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

\((1630,364)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{7}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1991 }(1730, a) \) \(1\)\(1\)\(e\left(\frac{17}{90}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{9}\right)\)
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)
 
\( \chi_{ 1991 }(1730,a) \;\) at \(\;a = \) e.g. 2