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Properties

Label 8100.463
Modulus $8100$
Conductor $8100$
Order $540$
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(8100, base_ring=CyclotomicField(540)) M = H._module chi = DirichletCharacter(H, M([270,380,513]))
 
Copy content gp:[g,chi] = znchar(Mod(463, 8100))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8100.463");
 

Basic properties

Modulus: \(8100\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8100\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(540\)
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 8100.do

\(\chi_{8100}(67,\cdot)\) \(\chi_{8100}(103,\cdot)\) \(\chi_{8100}(187,\cdot)\) \(\chi_{8100}(223,\cdot)\) \(\chi_{8100}(247,\cdot)\) \(\chi_{8100}(283,\cdot)\) \(\chi_{8100}(367,\cdot)\) \(\chi_{8100}(403,\cdot)\) \(\chi_{8100}(427,\cdot)\) \(\chi_{8100}(463,\cdot)\) \(\chi_{8100}(547,\cdot)\) \(\chi_{8100}(583,\cdot)\) \(\chi_{8100}(727,\cdot)\) \(\chi_{8100}(763,\cdot)\) \(\chi_{8100}(787,\cdot)\) \(\chi_{8100}(823,\cdot)\) \(\chi_{8100}(967,\cdot)\) \(\chi_{8100}(1003,\cdot)\) \(\chi_{8100}(1087,\cdot)\) \(\chi_{8100}(1123,\cdot)\) \(\chi_{8100}(1147,\cdot)\) \(\chi_{8100}(1183,\cdot)\) \(\chi_{8100}(1267,\cdot)\) \(\chi_{8100}(1303,\cdot)\) \(\chi_{8100}(1327,\cdot)\) \(\chi_{8100}(1363,\cdot)\) \(\chi_{8100}(1447,\cdot)\) \(\chi_{8100}(1483,\cdot)\)

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: $\Q(\zeta_{540})$
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 540 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((4051,6401,7777)\) → \((-1,e\left(\frac{19}{27}\right),e\left(\frac{19}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8100 }(463, a) \) \(1\)\(1\)\(e\left(\frac{55}{108}\right)\)\(e\left(\frac{229}{270}\right)\)\(e\left(\frac{367}{540}\right)\)\(e\left(\frac{103}{180}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{373}{540}\right)\)\(e\left(\frac{253}{270}\right)\)\(e\left(\frac{47}{270}\right)\)\(e\left(\frac{19}{180}\right)\)\(e\left(\frac{13}{135}\right)\)
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)
 
\( \chi_{ 8100 }(463,a) \;\) at \(\;a = \) e.g. 2