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Properties

Label 21489.8252
Modulus $21489$
Conductor $21489$
Order $126$
Real no
Primitive yes
Minimal yes
Parity odd

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(21489, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([63,105,98,18]))
 
Copy content gp:[g,chi] = znchar(Mod(8252, 21489))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("21489.8252");
 

Basic properties

Modulus: \(21489\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(21489\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(126\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 21489.uy

\(\chi_{21489}(23,\cdot)\) \(\chi_{21489}(719,\cdot)\) \(\chi_{21489}(1031,\cdot)\) \(\chi_{21489}(1499,\cdot)\) \(\chi_{21489}(2240,\cdot)\) \(\chi_{21489}(2981,\cdot)\) \(\chi_{21489}(3065,\cdot)\) \(\chi_{21489}(3728,\cdot)\) \(\chi_{21489}(3806,\cdot)\) \(\chi_{21489}(4424,\cdot)\) \(\chi_{21489}(4547,\cdot)\) \(\chi_{21489}(4664,\cdot)\) \(\chi_{21489}(5477,\cdot)\) \(\chi_{21489}(6686,\cdot)\) \(\chi_{21489}(8174,\cdot)\) \(\chi_{21489}(8252,\cdot)\) \(\chi_{21489}(8870,\cdot)\) \(\chi_{21489}(9851,\cdot)\) \(\chi_{21489}(10592,\cdot)\) \(\chi_{21489}(11132,\cdot)\) \(\chi_{21489}(11333,\cdot)\) \(\chi_{21489}(12146,\cdot)\) \(\chi_{21489}(12698,\cdot)\) \(\chi_{21489}(14843,\cdot)\) \(\chi_{21489}(15038,\cdot)\) \(\chi_{21489}(15539,\cdot)\) \(\chi_{21489}(17333,\cdot)\) \(\chi_{21489}(17801,\cdot)\) \(\chi_{21489}(18074,\cdot)\) \(\chi_{21489}(18815,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((14327,11572,2263,14821)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{7}{9}\right),e\left(\frac{1}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 21489 }(8252, a) \) \(-1\)\(1\)\(e\left(\frac{16}{63}\right)\)\(e\left(\frac{32}{63}\right)\)\(e\left(\frac{37}{63}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{53}{63}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{101}{126}\right)\)\(e\left(\frac{1}{63}\right)\)\(e\left(\frac{17}{18}\right)\)
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)
 
\( \chi_{ 21489 }(8252,a) \;\) at \(\;a = \) e.g. 2