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Properties

Label 4165.1187
Modulus $4165$
Conductor $4165$
Order $336$
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(336)) M = H._module chi = DirichletCharacter(H, M([84,320,189]))
 
Copy content gp:[g,chi] = znchar(Mod(1187, 4165))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4165.1187");
 

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: \(336\)
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.fh

\(\chi_{4165}(23,\cdot)\) \(\chi_{4165}(37,\cdot)\) \(\chi_{4165}(58,\cdot)\) \(\chi_{4165}(107,\cdot)\) \(\chi_{4165}(163,\cdot)\) \(\chi_{4165}(193,\cdot)\) \(\chi_{4165}(198,\cdot)\) \(\chi_{4165}(207,\cdot)\) \(\chi_{4165}(228,\cdot)\) \(\chi_{4165}(277,\cdot)\) \(\chi_{4165}(333,\cdot)\) \(\chi_{4165}(352,\cdot)\) \(\chi_{4165}(368,\cdot)\) \(\chi_{4165}(522,\cdot)\) \(\chi_{4165}(592,\cdot)\) \(\chi_{4165}(632,\cdot)\) \(\chi_{4165}(653,\cdot)\) \(\chi_{4165}(702,\cdot)\) \(\chi_{4165}(758,\cdot)\) \(\chi_{4165}(788,\cdot)\) \(\chi_{4165}(793,\cdot)\) \(\chi_{4165}(823,\cdot)\) \(\chi_{4165}(872,\cdot)\) \(\chi_{4165}(928,\cdot)\) \(\chi_{4165}(947,\cdot)\) \(\chi_{4165}(963,\cdot)\) \(\chi_{4165}(1017,\cdot)\) \(\chi_{4165}(1117,\cdot)\) \(\chi_{4165}(1187,\cdot)\) \(\chi_{4165}(1213,\cdot)\) ...

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: $\Q(\zeta_{336})$
Fixed field: Number field defined by a degree 336 polynomial (not computed)

Values on generators

\((1667,2551,2451)\) → \((i,e\left(\frac{20}{21}\right),e\left(\frac{9}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4165 }(1187, a) \) \(1\)\(1\)\(e\left(\frac{149}{168}\right)\)\(e\left(\frac{89}{336}\right)\)\(e\left(\frac{65}{84}\right)\)\(e\left(\frac{17}{112}\right)\)\(e\left(\frac{37}{56}\right)\)\(e\left(\frac{89}{168}\right)\)\(e\left(\frac{11}{336}\right)\)\(e\left(\frac{13}{336}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{23}{42}\right)\)
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)
 
\( \chi_{ 4165 }(1187,a) \;\) at \(\;a = \) e.g. 2