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Properties

Label 5184.4127
Modulus $5184$
Conductor $648$
Order $54$
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(5184, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([27,27,29]))
 
Copy content gp:[g,chi] = znchar(Mod(4127, 5184))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5184.4127");
 

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: \(648\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(54\)
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 \(\chi_{648}(563,\cdot)\)
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 5184.cd

\(\chi_{5184}(95,\cdot)\) \(\chi_{5184}(479,\cdot)\) \(\chi_{5184}(671,\cdot)\) \(\chi_{5184}(1055,\cdot)\) \(\chi_{5184}(1247,\cdot)\) \(\chi_{5184}(1631,\cdot)\) \(\chi_{5184}(1823,\cdot)\) \(\chi_{5184}(2207,\cdot)\) \(\chi_{5184}(2399,\cdot)\) \(\chi_{5184}(2783,\cdot)\) \(\chi_{5184}(2975,\cdot)\) \(\chi_{5184}(3359,\cdot)\) \(\chi_{5184}(3551,\cdot)\) \(\chi_{5184}(3935,\cdot)\) \(\chi_{5184}(4127,\cdot)\) \(\chi_{5184}(4511,\cdot)\) \(\chi_{5184}(4703,\cdot)\) \(\chi_{5184}(5087,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((2431,325,1217)\) → \((-1,-1,e\left(\frac{29}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 5184 }(4127, a) \) \(1\)\(1\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{5}{54}\right)\)\(e\left(\frac{53}{54}\right)\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{11}{27}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{13}{54}\right)\)
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)
 
\( \chi_{ 5184 }(4127,a) \;\) at \(\;a = \) e.g. 2