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Properties

Label 1580.907
Modulus $1580$
Conductor $1580$
Order $52$
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(1580, base_ring=CyclotomicField(52)) M = H._module chi = DirichletCharacter(H, M([26,13,24]))
 
Copy content gp:[g,chi] = znchar(Mod(907, 1580))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1580.907");
 

Basic properties

Modulus: \(1580\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1580\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(52\)
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 1580.bk

\(\chi_{1580}(67,\cdot)\) \(\chi_{1580}(87,\cdot)\) \(\chi_{1580}(143,\cdot)\) \(\chi_{1580}(223,\cdot)\) \(\chi_{1580}(247,\cdot)\) \(\chi_{1580}(283,\cdot)\) \(\chi_{1580}(383,\cdot)\) \(\chi_{1580}(403,\cdot)\) \(\chi_{1580}(447,\cdot)\) \(\chi_{1580}(563,\cdot)\) \(\chi_{1580}(763,\cdot)\) \(\chi_{1580}(887,\cdot)\) \(\chi_{1580}(907,\cdot)\) \(\chi_{1580}(1127,\cdot)\) \(\chi_{1580}(1203,\cdot)\) \(\chi_{1580}(1207,\cdot)\) \(\chi_{1580}(1223,\cdot)\) \(\chi_{1580}(1247,\cdot)\) \(\chi_{1580}(1407,\cdot)\) \(\chi_{1580}(1443,\cdot)\) \(\chi_{1580}(1487,\cdot)\) \(\chi_{1580}(1523,\cdot)\) \(\chi_{1580}(1547,\cdot)\) \(\chi_{1580}(1563,\cdot)\)

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: $\Q(\zeta_{52})$
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 52 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((791,317,161)\) → \((-1,i,e\left(\frac{6}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1580 }(907, a) \) \(1\)\(1\)\(e\left(\frac{37}{52}\right)\)\(e\left(\frac{11}{52}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{49}{52}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(i\)\(e\left(\frac{7}{52}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 1580 }(907,a) \;\) at \(\;a = \) e.g. 2