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Properties

Label 19961.21
Modulus $19961$
Conductor $19961$
Order $19960$
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(19961, base_ring=CyclotomicField(19960)) M = H._module chi = DirichletCharacter(H, M([13003]))
 
Copy content gp:[g,chi] = znchar(Mod(21, 19961))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("19961.21");
 

Basic properties

Modulus: \(19961\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(19961\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(19960\)
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 19961.p

\(\chi_{19961}(6,\cdot)\) \(\chi_{19961}(11,\cdot)\) \(\chi_{19961}(12,\cdot)\) \(\chi_{19961}(13,\cdot)\) \(\chi_{19961}(15,\cdot)\) \(\chi_{19961}(17,\cdot)\) \(\chi_{19961}(21,\cdot)\) \(\chi_{19961}(22,\cdot)\) \(\chi_{19961}(23,\cdot)\) \(\chi_{19961}(24,\cdot)\) \(\chi_{19961}(30,\cdot)\) \(\chi_{19961}(34,\cdot)\) \(\chi_{19961}(37,\cdot)\) \(\chi_{19961}(41,\cdot)\) \(\chi_{19961}(42,\cdot)\) \(\chi_{19961}(44,\cdot)\) \(\chi_{19961}(48,\cdot)\) \(\chi_{19961}(52,\cdot)\) \(\chi_{19961}(53,\cdot)\) \(\chi_{19961}(54,\cdot)\) \(\chi_{19961}(55,\cdot)\) \(\chi_{19961}(57,\cdot)\) \(\chi_{19961}(60,\cdot)\) \(\chi_{19961}(68,\cdot)\) \(\chi_{19961}(74,\cdot)\) \(\chi_{19961}(75,\cdot)\) \(\chi_{19961}(79,\cdot)\) \(\chi_{19961}(82,\cdot)\) \(\chi_{19961}(85,\cdot)\) \(\chi_{19961}(91,\cdot)\) ...

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: $\Q(\zeta_{19960})$
Fixed field: Number field defined by a degree 19960 polynomial (not computed)

Values on generators

\(6\) → \(e\left(\frac{13003}{19960}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 19961 }(21, a) \) \(-1\)\(1\)\(e\left(\frac{8059}{9980}\right)\)\(e\left(\frac{3369}{3992}\right)\)\(e\left(\frac{3069}{4990}\right)\)\(e\left(\frac{1056}{2495}\right)\)\(e\left(\frac{13003}{19960}\right)\)\(e\left(\frac{4981}{4990}\right)\)\(e\left(\frac{4217}{9980}\right)\)\(e\left(\frac{1373}{1996}\right)\)\(e\left(\frac{2303}{9980}\right)\)\(e\left(\frac{15621}{19960}\right)\)
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)
 
\( \chi_{ 19961 }(21,a) \;\) at \(\;a = \) e.g. 2