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Properties

Label 549261.1321
Modulus $549261$
Conductor $549261$
Order $30510$
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(549261, base_ring=CyclotomicField(30510)) M = H._module chi = DirichletCharacter(H, M([25990,9819]))
 
Copy content gp:[g,chi] = znchar(Mod(1321, 549261))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("549261.1321");
 

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: \(549261\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(30510\)
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 549261.jx

\(\chi_{549261}(4,\cdot)\) \(\chi_{549261}(76,\cdot)\) \(\chi_{549261}(94,\cdot)\) \(\chi_{549261}(130,\cdot)\) \(\chi_{549261}(157,\cdot)\) \(\chi_{549261}(166,\cdot)\) \(\chi_{549261}(184,\cdot)\) \(\chi_{549261}(463,\cdot)\) \(\chi_{549261}(484,\cdot)\) \(\chi_{549261}(529,\cdot)\) \(\chi_{549261}(580,\cdot)\) \(\chi_{549261}(607,\cdot)\) \(\chi_{549261}(610,\cdot)\) \(\chi_{549261}(997,\cdot)\) \(\chi_{549261}(1156,\cdot)\) \(\chi_{549261}(1282,\cdot)\) \(\chi_{549261}(1321,\cdot)\) \(\chi_{549261}(1444,\cdot)\) \(\chi_{549261}(1447,\cdot)\) \(\chi_{549261}(1471,\cdot)\)

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: $\Q(\zeta_{15255})$
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 30510 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((47468,501796)\) → \((e\left(\frac{23}{27}\right),e\left(\frac{1091}{3390}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 549261 }(1321, a) \) \(1\)\(1\)\(e\left(\frac{5299}{30510}\right)\)\(e\left(\frac{5299}{15255}\right)\)\(e\left(\frac{10858}{15255}\right)\)\(e\left(\frac{20083}{30510}\right)\)\(e\left(\frac{5299}{10170}\right)\)\(e\left(\frac{1801}{2034}\right)\)\(e\left(\frac{2084}{15255}\right)\)\(e\left(\frac{8417}{30510}\right)\)\(e\left(\frac{12691}{15255}\right)\)\(e\left(\frac{10598}{15255}\right)\)
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)
 
\( \chi_{ 549261 }(1321,a) \;\) at \(\;a = \) e.g. 2