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Properties

Label 18525.2182
Modulus $18525$
Conductor $1235$
Order $36$
Real no
Primitive no
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(18525, base_ring=CyclotomicField(36)) M = H._module chi = DirichletCharacter(H, M([0,9,21,8]))
 
Copy content gp:[g,chi] = znchar(Mod(2182, 18525))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("18525.2182");
 

Basic properties

Modulus: \(18525\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1235\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(36\)
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: no, induced from \(\chi_{1235}(947,\cdot)\)
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 18525.qg

\(\chi_{18525}(2182,\cdot)\) \(\chi_{18525}(2593,\cdot)\) \(\chi_{18525}(2932,\cdot)\) \(\chi_{18525}(4318,\cdot)\) \(\chi_{18525}(6268,\cdot)\) \(\chi_{18525}(8782,\cdot)\) \(\chi_{18525}(9193,\cdot)\) \(\chi_{18525}(11368,\cdot)\) \(\chi_{18525}(12682,\cdot)\) \(\chi_{18525}(15832,\cdot)\) \(\chi_{18525}(17218,\cdot)\) \(\chi_{18525}(17782,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((6176,8152,7126,15601)\) → \((1,i,e\left(\frac{7}{12}\right),e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(22\)\(23\)
\( \chi_{ 18525 }(2182, a) \) \(1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(1\)\(e\left(\frac{1}{6}\right)\)\(-i\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{1}{36}\right)\)
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)
 
\( \chi_{ 18525 }(2182,a) \;\) at \(\;a = \) e.g. 2