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Properties

Label 1637.25
Modulus $1637$
Conductor $1637$
Order $818$
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(1637, base_ring=CyclotomicField(818)) M = H._module chi = DirichletCharacter(H, M([113]))
 
Copy content gp:[g,chi] = znchar(Mod(25, 1637))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1637.25");
 

Basic properties

Modulus: \(1637\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1637\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(818\)
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 1637.e

\(\chi_{1637}(4,\cdot)\) \(\chi_{1637}(6,\cdot)\) \(\chi_{1637}(9,\cdot)\) \(\chi_{1637}(14,\cdot)\) \(\chi_{1637}(21,\cdot)\) \(\chi_{1637}(25,\cdot)\) \(\chi_{1637}(29,\cdot)\) \(\chi_{1637}(31,\cdot)\) \(\chi_{1637}(37,\cdot)\) \(\chi_{1637}(38,\cdot)\) \(\chi_{1637}(40,\cdot)\) \(\chi_{1637}(44,\cdot)\) \(\chi_{1637}(49,\cdot)\) \(\chi_{1637}(52,\cdot)\) \(\chi_{1637}(57,\cdot)\) \(\chi_{1637}(60,\cdot)\) \(\chi_{1637}(64,\cdot)\) \(\chi_{1637}(66,\cdot)\) \(\chi_{1637}(78,\cdot)\) \(\chi_{1637}(85,\cdot)\) \(\chi_{1637}(86,\cdot)\) \(\chi_{1637}(90,\cdot)\) \(\chi_{1637}(92,\cdot)\) \(\chi_{1637}(96,\cdot)\) \(\chi_{1637}(99,\cdot)\) \(\chi_{1637}(117,\cdot)\) \(\chi_{1637}(118,\cdot)\) \(\chi_{1637}(122,\cdot)\) \(\chi_{1637}(127,\cdot)\) \(\chi_{1637}(129,\cdot)\) ...

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: $\Q(\zeta_{409})$
Fixed field: Number field defined by a degree 818 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{113}{818}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1637 }(25, a) \) \(1\)\(1\)\(e\left(\frac{113}{818}\right)\)\(e\left(\frac{169}{818}\right)\)\(e\left(\frac{113}{409}\right)\)\(e\left(\frac{499}{818}\right)\)\(e\left(\frac{141}{409}\right)\)\(e\left(\frac{651}{818}\right)\)\(e\left(\frac{339}{818}\right)\)\(e\left(\frac{169}{409}\right)\)\(e\left(\frac{306}{409}\right)\)\(e\left(\frac{36}{409}\right)\)
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)
 
\( \chi_{ 1637 }(25,a) \;\) at \(\;a = \) e.g. 2