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Properties

Label 549261.47468
Modulus $549261$
Conductor $81$
Order $54$
Real no
Primitive no
Minimal yes
Parity odd

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(549261, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([1,0]))
 
Copy content gp:[g,chi] = znchar(Mod(47468, 549261))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("549261.47468");
 

Basic properties

Modulus: \(549261\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(81\)
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 \(\chi_{81}(2,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 549261.dd

\(\chi_{549261}(6782,\cdot)\) \(\chi_{549261}(47468,\cdot)\) \(\chi_{549261}(67811,\cdot)\) \(\chi_{549261}(108497,\cdot)\) \(\chi_{549261}(128840,\cdot)\) \(\chi_{549261}(169526,\cdot)\) \(\chi_{549261}(189869,\cdot)\) \(\chi_{549261}(230555,\cdot)\) \(\chi_{549261}(250898,\cdot)\) \(\chi_{549261}(291584,\cdot)\) \(\chi_{549261}(311927,\cdot)\) \(\chi_{549261}(352613,\cdot)\) \(\chi_{549261}(372956,\cdot)\) \(\chi_{549261}(413642,\cdot)\) \(\chi_{549261}(433985,\cdot)\) \(\chi_{549261}(474671,\cdot)\) \(\chi_{549261}(495014,\cdot)\) \(\chi_{549261}(535700,\cdot)\)

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: \(\Q(\zeta_{27})\)
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 54 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((47468,501796)\) → \((e\left(\frac{1}{54}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 549261 }(47468, a) \) \(-1\)\(1\)\(e\left(\frac{1}{54}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{8}{27}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{13}{54}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{17}{54}\right)\)\(e\left(\frac{2}{27}\right)\)
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)
 
\( \chi_{ 549261 }(47468,a) \;\) at \(\;a = \) e.g. 2