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Properties

Label 4165.2112
Modulus $4165$
Conductor $4165$
Order $84$
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(4165, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([21,58,63]))
 
Copy content gp:[g,chi] = znchar(Mod(2112, 4165))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4165.2112");
 

Basic properties

Modulus: \(4165\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4165\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 4165.en

\(\chi_{4165}(157,\cdot)\) \(\chi_{4165}(327,\cdot)\) \(\chi_{4165}(353,\cdot)\) \(\chi_{4165}(523,\cdot)\) \(\chi_{4165}(752,\cdot)\) \(\chi_{4165}(922,\cdot)\) \(\chi_{4165}(948,\cdot)\) \(\chi_{4165}(1118,\cdot)\) \(\chi_{4165}(1347,\cdot)\) \(\chi_{4165}(1517,\cdot)\) \(\chi_{4165}(1543,\cdot)\) \(\chi_{4165}(1713,\cdot)\) \(\chi_{4165}(2112,\cdot)\) \(\chi_{4165}(2308,\cdot)\) \(\chi_{4165}(2537,\cdot)\) \(\chi_{4165}(2707,\cdot)\) \(\chi_{4165}(2733,\cdot)\) \(\chi_{4165}(2903,\cdot)\) \(\chi_{4165}(3132,\cdot)\) \(\chi_{4165}(3328,\cdot)\) \(\chi_{4165}(3727,\cdot)\) \(\chi_{4165}(3897,\cdot)\) \(\chi_{4165}(3923,\cdot)\) \(\chi_{4165}(4093,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1667,2551,2451)\) → \((i,e\left(\frac{29}{42}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4165 }(2112, a) \) \(1\)\(1\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{17}{21}\right)\)
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)
 
\( \chi_{ 4165 }(2112,a) \;\) at \(\;a = \) e.g. 2