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Properties

Label 541.185
Modulus $541$
Conductor $541$
Order $108$
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(541, base_ring=CyclotomicField(108)) M = H._module chi = DirichletCharacter(H, M([31]))
 
Copy content gp:[g,chi] = znchar(Mod(185, 541))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("541.185");
 

Basic properties

Modulus: \(541\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(541\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(108\)
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 541.t

\(\chi_{541}(11,\cdot)\) \(\chi_{541}(32,\cdot)\) \(\chi_{541}(61,\cdot)\) \(\chi_{541}(70,\cdot)\) \(\chi_{541}(85,\cdot)\) \(\chi_{541}(97,\cdot)\) \(\chi_{541}(145,\cdot)\) \(\chi_{541}(165,\cdot)\) \(\chi_{541}(167,\cdot)\) \(\chi_{541}(168,\cdot)\) \(\chi_{541}(185,\cdot)\) \(\chi_{541}(186,\cdot)\) \(\chi_{541}(190,\cdot)\) \(\chi_{541}(193,\cdot)\) \(\chi_{541}(200,\cdot)\) \(\chi_{541}(204,\cdot)\) \(\chi_{541}(230,\cdot)\) \(\chi_{541}(246,\cdot)\) \(\chi_{541}(295,\cdot)\) \(\chi_{541}(311,\cdot)\) \(\chi_{541}(337,\cdot)\) \(\chi_{541}(341,\cdot)\) \(\chi_{541}(348,\cdot)\) \(\chi_{541}(351,\cdot)\) \(\chi_{541}(355,\cdot)\) \(\chi_{541}(356,\cdot)\) \(\chi_{541}(373,\cdot)\) \(\chi_{541}(374,\cdot)\) \(\chi_{541}(376,\cdot)\) \(\chi_{541}(396,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{31}{108}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 541 }(185, a) \) \(-1\)\(1\)\(e\left(\frac{31}{108}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{31}{54}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{71}{108}\right)\)\(e\left(\frac{73}{108}\right)\)
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)
 
\( \chi_{ 541 }(185,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content comment:Gauss sum
 
Copy content sage:chi.gauss_sum(a)
 
Copy content gp:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 541 }(185,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 541 }(185,·),\chi_{ 541 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 541 }(185,·)) \;\) at \(\; a,b = \) e.g. 1,2