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Properties

Label 5184.815
Modulus $5184$
Conductor $1296$
Order $108$
Real no
Primitive no
Minimal no
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(5184, base_ring=CyclotomicField(108)) M = H._module chi = DirichletCharacter(H, M([54,81,46]))
 
Copy content gp:[g,chi] = znchar(Mod(815, 5184))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5184.815");
 

Basic properties

Modulus: \(5184\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1296\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(108\)
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 \(\chi_{1296}(1139,\cdot)\)
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: no
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 5184.cm

\(\chi_{5184}(47,\cdot)\) \(\chi_{5184}(239,\cdot)\) \(\chi_{5184}(335,\cdot)\) \(\chi_{5184}(527,\cdot)\) \(\chi_{5184}(623,\cdot)\) \(\chi_{5184}(815,\cdot)\) \(\chi_{5184}(911,\cdot)\) \(\chi_{5184}(1103,\cdot)\) \(\chi_{5184}(1199,\cdot)\) \(\chi_{5184}(1391,\cdot)\) \(\chi_{5184}(1487,\cdot)\) \(\chi_{5184}(1679,\cdot)\) \(\chi_{5184}(1775,\cdot)\) \(\chi_{5184}(1967,\cdot)\) \(\chi_{5184}(2063,\cdot)\) \(\chi_{5184}(2255,\cdot)\) \(\chi_{5184}(2351,\cdot)\) \(\chi_{5184}(2543,\cdot)\) \(\chi_{5184}(2639,\cdot)\) \(\chi_{5184}(2831,\cdot)\) \(\chi_{5184}(2927,\cdot)\) \(\chi_{5184}(3119,\cdot)\) \(\chi_{5184}(3215,\cdot)\) \(\chi_{5184}(3407,\cdot)\) \(\chi_{5184}(3503,\cdot)\) \(\chi_{5184}(3695,\cdot)\) \(\chi_{5184}(3791,\cdot)\) \(\chi_{5184}(3983,\cdot)\) \(\chi_{5184}(4079,\cdot)\) \(\chi_{5184}(4271,\cdot)\) ...

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: $\Q(\zeta_{108})$
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 108 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((2431,325,1217)\) → \((-1,-i,e\left(\frac{23}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 5184 }(815, a) \) \(1\)\(1\)\(e\left(\frac{59}{108}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{85}{108}\right)\)\(e\left(\frac{71}{108}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{5}{54}\right)\)\(e\left(\frac{1}{108}\right)\)\(e\left(\frac{1}{54}\right)\)
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)
 
\( \chi_{ 5184 }(815,a) \;\) at \(\;a = \) e.g. 2