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Properties

Label 3751.541
Modulus $3751$
Conductor $3751$
Order $330$
Real no
Primitive yes
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(3751, base_ring=CyclotomicField(330)) M = H._module chi = DirichletCharacter(H, M([183,242]))
 
Copy content gp:[g,chi] = znchar(Mod(541, 3751))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.541");
 

Basic properties

Modulus: \(3751\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3751\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(330\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3751.dx

\(\chi_{3751}(173,\cdot)\) \(\chi_{3751}(193,\cdot)\) \(\chi_{3751}(200,\cdot)\) \(\chi_{3751}(205,\cdot)\) \(\chi_{3751}(227,\cdot)\) \(\chi_{3751}(237,\cdot)\) \(\chi_{3751}(288,\cdot)\) \(\chi_{3751}(338,\cdot)\) \(\chi_{3751}(514,\cdot)\) \(\chi_{3751}(534,\cdot)\) \(\chi_{3751}(541,\cdot)\) \(\chi_{3751}(546,\cdot)\) \(\chi_{3751}(568,\cdot)\) \(\chi_{3751}(629,\cdot)\) \(\chi_{3751}(679,\cdot)\) \(\chi_{3751}(855,\cdot)\) \(\chi_{3751}(875,\cdot)\) \(\chi_{3751}(882,\cdot)\) \(\chi_{3751}(909,\cdot)\) \(\chi_{3751}(919,\cdot)\) \(\chi_{3751}(970,\cdot)\) \(\chi_{3751}(1020,\cdot)\) \(\chi_{3751}(1196,\cdot)\) \(\chi_{3751}(1216,\cdot)\) \(\chi_{3751}(1223,\cdot)\) \(\chi_{3751}(1228,\cdot)\) \(\chi_{3751}(1260,\cdot)\) \(\chi_{3751}(1311,\cdot)\) \(\chi_{3751}(1361,\cdot)\) \(\chi_{3751}(1537,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((2543,2421)\) → \((e\left(\frac{61}{110}\right),e\left(\frac{11}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 3751 }(541, a) \) \(-1\)\(1\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{116}{165}\right)\)\(e\left(\frac{227}{330}\right)\)\(e\left(\frac{137}{330}\right)\)\(e\left(\frac{51}{110}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{283}{330}\right)\)\(e\left(\frac{139}{165}\right)\)
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)
 
\( \chi_{ 3751 }(541,a) \;\) at \(\;a = \) e.g. 2