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Properties

Label 2263.2062
Modulus $2263$
Conductor $2263$
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(2263, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([18,25]))
 
Copy content gp:[g,chi] = znchar(Mod(2062, 2263))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2263.2062");
 

Basic properties

Modulus: \(2263\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2263\)
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 2263.eb

\(\chi_{2263}(109,\cdot)\) \(\chi_{2263}(349,\cdot)\) \(\chi_{2263}(436,\cdot)\) \(\chi_{2263}(529,\cdot)\) \(\chi_{2263}(653,\cdot)\) \(\chi_{2263}(655,\cdot)\) \(\chi_{2263}(698,\cdot)\) \(\chi_{2263}(748,\cdot)\) \(\chi_{2263}(839,\cdot)\) \(\chi_{2263}(872,\cdot)\) \(\chi_{2263}(1058,\cdot)\) \(\chi_{2263}(1093,\cdot)\) \(\chi_{2263}(1186,\cdot)\) \(\chi_{2263}(1225,\cdot)\) \(\chi_{2263}(1310,\cdot)\) \(\chi_{2263}(1428,\cdot)\) \(\chi_{2263}(1496,\cdot)\) \(\chi_{2263}(1647,\cdot)\) \(\chi_{2263}(1955,\cdot)\) \(\chi_{2263}(1969,\cdot)\) \(\chi_{2263}(2062,\cdot)\) \(\chi_{2263}(2085,\cdot)\) \(\chi_{2263}(2174,\cdot)\) \(\chi_{2263}(2186,\cdot)\)

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

\((220,1830)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2263 }(2062, a) \) \(1\)\(1\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{79}{90}\right)\)
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)
 
\( \chi_{ 2263 }(2062,a) \;\) at \(\;a = \) e.g. 2