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Properties

Label 5415.382
Modulus $5415$
Conductor $1805$
Order $684$
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(5415, base_ring=CyclotomicField(684)) M = H._module chi = DirichletCharacter(H, M([0,171,578]))
 
Copy content gp:[g,chi] = znchar(Mod(382, 5415))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5415.382");
 

Basic properties

Modulus: \(5415\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1805\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(684\)
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 \(\chi_{1805}(382,\cdot)\)
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 5415.cs

\(\chi_{5415}(13,\cdot)\) \(\chi_{5415}(22,\cdot)\) \(\chi_{5415}(52,\cdot)\) \(\chi_{5415}(67,\cdot)\) \(\chi_{5415}(97,\cdot)\) \(\chi_{5415}(148,\cdot)\) \(\chi_{5415}(193,\cdot)\) \(\chi_{5415}(223,\cdot)\) \(\chi_{5415}(238,\cdot)\) \(\chi_{5415}(268,\cdot)\) \(\chi_{5415}(298,\cdot)\) \(\chi_{5415}(337,\cdot)\) \(\chi_{5415}(352,\cdot)\) \(\chi_{5415}(382,\cdot)\) \(\chi_{5415}(412,\cdot)\) \(\chi_{5415}(433,\cdot)\) \(\chi_{5415}(478,\cdot)\) \(\chi_{5415}(508,\cdot)\) \(\chi_{5415}(523,\cdot)\) \(\chi_{5415}(547,\cdot)\) \(\chi_{5415}(553,\cdot)\) \(\chi_{5415}(583,\cdot)\) \(\chi_{5415}(592,\cdot)\) \(\chi_{5415}(622,\cdot)\) \(\chi_{5415}(637,\cdot)\) \(\chi_{5415}(667,\cdot)\) \(\chi_{5415}(697,\cdot)\) \(\chi_{5415}(718,\cdot)\) \(\chi_{5415}(763,\cdot)\) \(\chi_{5415}(793,\cdot)\) ...

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: $\Q(\zeta_{684})$
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 684 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((3611,2167,5056)\) → \((1,i,e\left(\frac{289}{342}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(22\)
\( \chi_{ 5415 }(382, a) \) \(1\)\(1\)\(e\left(\frac{65}{684}\right)\)\(e\left(\frac{65}{342}\right)\)\(e\left(\frac{1}{228}\right)\)\(e\left(\frac{65}{228}\right)\)\(e\left(\frac{11}{57}\right)\)\(e\left(\frac{523}{684}\right)\)\(e\left(\frac{17}{171}\right)\)\(e\left(\frac{65}{171}\right)\)\(e\left(\frac{155}{684}\right)\)\(e\left(\frac{197}{684}\right)\)
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)
 
\( \chi_{ 5415 }(382,a) \;\) at \(\;a = \) e.g. 2