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Properties

Label 36667.1018
Modulus $36667$
Conductor $36667$
Order $1980$
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(36667, base_ring=CyclotomicField(1980)) M = H._module chi = DirichletCharacter(H, M([1925,1314]))
 
Copy content gp:[g,chi] = znchar(Mod(1018, 36667))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("36667.1018");
 

Basic properties

Modulus: \(36667\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(36667\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1980\)
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 36667.oz

\(\chi_{36667}(35,\cdot)\) \(\chi_{36667}(87,\cdot)\) \(\chi_{36667}(143,\cdot)\) \(\chi_{36667}(217,\cdot)\) \(\chi_{36667}(298,\cdot)\) \(\chi_{36667}(372,\cdot)\) \(\chi_{36667}(442,\cdot)\) \(\chi_{36667}(479,\cdot)\) \(\chi_{36667}(538,\cdot)\) \(\chi_{36667}(624,\cdot)\) \(\chi_{36667}(681,\cdot)\) \(\chi_{36667}(757,\cdot)\) \(\chi_{36667}(759,\cdot)\) \(\chi_{36667}(762,\cdot)\) \(\chi_{36667}(940,\cdot)\) \(\chi_{36667}(1018,\cdot)\) \(\chi_{36667}(1051,\cdot)\) \(\chi_{36667}(1078,\cdot)\) \(\chi_{36667}(1127,\cdot)\) \(\chi_{36667}(1134,\cdot)\) \(\chi_{36667}(1186,\cdot)\) \(\chi_{36667}(1204,\cdot)\) \(\chi_{36667}(1208,\cdot)\) \(\chi_{36667}(1253,\cdot)\) \(\chi_{36667}(1330,\cdot)\) \(\chi_{36667}(1347,\cdot)\) \(\chi_{36667}(1350,\cdot)\) \(\chi_{36667}(1462,\cdot)\)

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: $\Q(\zeta_{1980})$
Fixed field: Number field defined by a degree 1980 polynomial (not computed)

Values on generators

\((22794,32709)\) → \((e\left(\frac{35}{36}\right),e\left(\frac{73}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 36667 }(1018, a) \) \(1\)\(1\)\(e\left(\frac{593}{1980}\right)\)\(e\left(\frac{304}{495}\right)\)\(e\left(\frac{593}{990}\right)\)\(e\left(\frac{1831}{1980}\right)\)\(e\left(\frac{201}{220}\right)\)\(e\left(\frac{713}{990}\right)\)\(e\left(\frac{593}{660}\right)\)\(e\left(\frac{113}{495}\right)\)\(e\left(\frac{37}{165}\right)\)\(e\left(\frac{17}{165}\right)\)
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)
 
\( \chi_{ 36667 }(1018,a) \;\) at \(\;a = \) e.g. 2