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Properties

Label 1609.39
Modulus $1609$
Conductor $1609$
Order $804$
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(1609, base_ring=CyclotomicField(804)) M = H._module chi = DirichletCharacter(H, M([31]))
 
Copy content gp:[g,chi] = znchar(Mod(39, 1609))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1609.39");
 

Basic properties

Modulus: \(1609\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1609\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(804\)
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 1609.o

\(\chi_{1609}(6,\cdot)\) \(\chi_{1609}(12,\cdot)\) \(\chi_{1609}(30,\cdot)\) \(\chi_{1609}(33,\cdot)\) \(\chi_{1609}(39,\cdot)\) \(\chi_{1609}(48,\cdot)\) \(\chi_{1609}(49,\cdot)\) \(\chi_{1609}(54,\cdot)\) \(\chi_{1609}(59,\cdot)\) \(\chi_{1609}(60,\cdot)\) \(\chi_{1609}(62,\cdot)\) \(\chi_{1609}(78,\cdot)\) \(\chi_{1609}(82,\cdot)\) \(\chi_{1609}(96,\cdot)\) \(\chi_{1609}(98,\cdot)\) \(\chi_{1609}(107,\cdot)\) \(\chi_{1609}(108,\cdot)\) \(\chi_{1609}(109,\cdot)\) \(\chi_{1609}(118,\cdot)\) \(\chi_{1609}(124,\cdot)\) \(\chi_{1609}(132,\cdot)\) \(\chi_{1609}(133,\cdot)\) \(\chi_{1609}(134,\cdot)\) \(\chi_{1609}(146,\cdot)\) \(\chi_{1609}(149,\cdot)\) \(\chi_{1609}(150,\cdot)\) \(\chi_{1609}(164,\cdot)\) \(\chi_{1609}(165,\cdot)\) \(\chi_{1609}(195,\cdot)\) \(\chi_{1609}(197,\cdot)\) ...

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: $\Q(\zeta_{804})$
Fixed field: Number field defined by a degree 804 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{31}{804}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1609 }(39, a) \) \(1\)\(1\)\(e\left(\frac{35}{201}\right)\)\(e\left(\frac{25}{134}\right)\)\(e\left(\frac{70}{201}\right)\)\(e\left(\frac{11}{67}\right)\)\(e\left(\frac{145}{402}\right)\)\(e\left(\frac{31}{804}\right)\)\(e\left(\frac{35}{67}\right)\)\(e\left(\frac{25}{67}\right)\)\(e\left(\frac{68}{201}\right)\)\(e\left(\frac{52}{201}\right)\)
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)
 
\( \chi_{ 1609 }(39,a) \;\) at \(\;a = \) e.g. 2