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Properties

Label 1000.809
Modulus $1000$
Conductor $125$
Order $50$
Real no
Primitive no
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(1000, base_ring=CyclotomicField(50)) M = H._module chi = DirichletCharacter(H, M([0,0,17]))
 
Copy content gp:[g,chi] = znchar(Mod(809, 1000))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1000.809");
 

Basic properties

Modulus: \(1000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(125\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(50\)
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: no, induced from \(\chi_{125}(59,\cdot)\)
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 1000.be

\(\chi_{1000}(9,\cdot)\) \(\chi_{1000}(89,\cdot)\) \(\chi_{1000}(129,\cdot)\) \(\chi_{1000}(169,\cdot)\) \(\chi_{1000}(209,\cdot)\) \(\chi_{1000}(289,\cdot)\) \(\chi_{1000}(329,\cdot)\) \(\chi_{1000}(369,\cdot)\) \(\chi_{1000}(409,\cdot)\) \(\chi_{1000}(489,\cdot)\) \(\chi_{1000}(529,\cdot)\) \(\chi_{1000}(569,\cdot)\) \(\chi_{1000}(609,\cdot)\) \(\chi_{1000}(689,\cdot)\) \(\chi_{1000}(729,\cdot)\) \(\chi_{1000}(769,\cdot)\) \(\chi_{1000}(809,\cdot)\) \(\chi_{1000}(889,\cdot)\) \(\chi_{1000}(929,\cdot)\) \(\chi_{1000}(969,\cdot)\)

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: \(\Q(\zeta_{25})\)
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 50 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((751,501,377)\) → \((1,1,e\left(\frac{17}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1000 }(809, a) \) \(1\)\(1\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{19}{25}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{27}{50}\right)\)\(e\left(\frac{7}{50}\right)\)
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)
 
\( \chi_{ 1000 }(809,a) \;\) at \(\;a = \) e.g. 2