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Properties

Label 416000.bth
Modulus $416000$
Conductor $416000$
Order $4800$
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 orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416000, base_ring=CyclotomicField(4800)) M = H._module chi = DirichletCharacter(H, M([0,4425,2976,1600])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(29, 416000)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416000.29"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(416000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(416000\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(4800\)
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);
 

Related number fields

Field of values: $\Q(\zeta_{4800})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 4800 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

First 5 of 1280 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(17\) \(19\) \(21\) \(23\) \(27\) \(29\)
\(\chi_{416000}(29,\cdot)\) \(1\) \(1\) \(e\left(\frac{4507}{4800}\right)\) \(e\left(\frac{281}{480}\right)\) \(e\left(\frac{2107}{2400}\right)\) \(e\left(\frac{3901}{4800}\right)\) \(e\left(\frac{887}{1200}\right)\) \(e\left(\frac{143}{4800}\right)\) \(e\left(\frac{839}{1600}\right)\) \(e\left(\frac{1103}{2400}\right)\) \(e\left(\frac{1307}{1600}\right)\) \(e\left(\frac{787}{4800}\right)\)
\(\chi_{416000}(269,\cdot)\) \(1\) \(1\) \(e\left(\frac{3023}{4800}\right)\) \(e\left(\frac{469}{480}\right)\) \(e\left(\frac{623}{2400}\right)\) \(e\left(\frac{3689}{4800}\right)\) \(e\left(\frac{43}{1200}\right)\) \(e\left(\frac{2227}{4800}\right)\) \(e\left(\frac{971}{1600}\right)\) \(e\left(\frac{1267}{2400}\right)\) \(e\left(\frac{1423}{1600}\right)\) \(e\left(\frac{743}{4800}\right)\)
\(\chi_{416000}(789,\cdot)\) \(1\) \(1\) \(e\left(\frac{2381}{4800}\right)\) \(e\left(\frac{223}{480}\right)\) \(e\left(\frac{2381}{2400}\right)\) \(e\left(\frac{1883}{4800}\right)\) \(e\left(\frac{721}{1200}\right)\) \(e\left(\frac{3769}{4800}\right)\) \(e\left(\frac{1537}{1600}\right)\) \(e\left(\frac{1849}{2400}\right)\) \(e\left(\frac{781}{1600}\right)\) \(e\left(\frac{3221}{4800}\right)\)
\(\chi_{416000}(1069,\cdot)\) \(1\) \(1\) \(e\left(\frac{3943}{4800}\right)\) \(e\left(\frac{29}{480}\right)\) \(e\left(\frac{1543}{2400}\right)\) \(e\left(\frac{2449}{4800}\right)\) \(e\left(\frac{563}{1200}\right)\) \(e\left(\frac{107}{4800}\right)\) \(e\left(\frac{1411}{1600}\right)\) \(e\left(\frac{1547}{2400}\right)\) \(e\left(\frac{743}{1600}\right)\) \(e\left(\frac{1663}{4800}\right)\)
\(\chi_{416000}(1309,\cdot)\) \(1\) \(1\) \(e\left(\frac{1499}{4800}\right)\) \(e\left(\frac{217}{480}\right)\) \(e\left(\frac{1499}{2400}\right)\) \(e\left(\frac{4157}{4800}\right)\) \(e\left(\frac{1159}{1200}\right)\) \(e\left(\frac{3151}{4800}\right)\) \(e\left(\frac{1223}{1600}\right)\) \(e\left(\frac{271}{2400}\right)\) \(e\left(\frac{1499}{1600}\right)\) \(e\left(\frac{659}{4800}\right)\)