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Properties

Label 3823.757
Modulus $3823$
Conductor $3823$
Order $546$
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(3823, base_ring=CyclotomicField(546)) M = H._module chi = DirichletCharacter(H, M([491]))
 
Copy content gp:[g,chi] = znchar(Mod(757, 3823))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3823.757");
 

Basic properties

Modulus: \(3823\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3823\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(546\)
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 3823.t

\(\chi_{3823}(12,\cdot)\) \(\chi_{3823}(33,\cdot)\) \(\chi_{3823}(79,\cdot)\) \(\chi_{3823}(91,\cdot)\) \(\chi_{3823}(102,\cdot)\) \(\chi_{3823}(147,\cdot)\) \(\chi_{3823}(159,\cdot)\) \(\chi_{3823}(179,\cdot)\) \(\chi_{3823}(180,\cdot)\) \(\chi_{3823}(188,\cdot)\) \(\chi_{3823}(211,\cdot)\) \(\chi_{3823}(226,\cdot)\) \(\chi_{3823}(247,\cdot)\) \(\chi_{3823}(321,\cdot)\) \(\chi_{3823}(337,\cdot)\) \(\chi_{3823}(401,\cdot)\) \(\chi_{3823}(413,\cdot)\) \(\chi_{3823}(508,\cdot)\) \(\chi_{3823}(517,\cdot)\) \(\chi_{3823}(567,\cdot)\) \(\chi_{3823}(647,\cdot)\) \(\chi_{3823}(683,\cdot)\) \(\chi_{3823}(692,\cdot)\) \(\chi_{3823}(732,\cdot)\) \(\chi_{3823}(757,\cdot)\) \(\chi_{3823}(788,\cdot)\) \(\chi_{3823}(817,\cdot)\) \(\chi_{3823}(823,\cdot)\) \(\chi_{3823}(831,\cdot)\) \(\chi_{3823}(842,\cdot)\) ...

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: $\Q(\zeta_{273})$
Fixed field: Number field defined by a degree 546 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{491}{546}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3823 }(757, a) \) \(-1\)\(1\)\(e\left(\frac{25}{91}\right)\)\(e\left(\frac{491}{546}\right)\)\(e\left(\frac{50}{91}\right)\)\(e\left(\frac{23}{182}\right)\)\(e\left(\frac{95}{546}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{75}{91}\right)\)\(e\left(\frac{218}{273}\right)\)\(e\left(\frac{73}{182}\right)\)\(e\left(\frac{22}{91}\right)\)
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)
 
\( \chi_{ 3823 }(757,a) \;\) at \(\;a = \) e.g. 2