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Properties

Label 35539.887
Modulus $35539$
Conductor $35539$
Order $2538$
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(35539, base_ring=CyclotomicField(2538)) M = H._module chi = DirichletCharacter(H, M([2115,1877]))
 
Copy content gp:[g,chi] = znchar(Mod(887, 35539))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35539.887");
 

Basic properties

Modulus: \(35539\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(35539\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2538\)
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 35539.et

\(\chi_{35539}(47,\cdot)\) \(\chi_{35539}(61,\cdot)\) \(\chi_{35539}(138,\cdot)\) \(\chi_{35539}(171,\cdot)\) \(\chi_{35539}(388,\cdot)\) \(\chi_{35539}(432,\cdot)\) \(\chi_{35539}(542,\cdot)\) \(\chi_{35539}(572,\cdot)\) \(\chi_{35539}(647,\cdot)\) \(\chi_{35539}(663,\cdot)\) \(\chi_{35539}(724,\cdot)\) \(\chi_{35539}(740,\cdot)\) \(\chi_{35539}(752,\cdot)\) \(\chi_{35539}(759,\cdot)\) \(\chi_{35539}(796,\cdot)\) \(\chi_{35539}(810,\cdot)\) \(\chi_{35539}(866,\cdot)\) \(\chi_{35539}(871,\cdot)\) \(\chi_{35539}(887,\cdot)\) \(\chi_{35539}(927,\cdot)\) \(\chi_{35539}(943,\cdot)\) \(\chi_{35539}(976,\cdot)\) \(\chi_{35539}(997,\cdot)\) \(\chi_{35539}(1041,\cdot)\) \(\chi_{35539}(1083,\cdot)\) \(\chi_{35539}(1111,\cdot)\) \(\chi_{35539}(1235,\cdot)\) \(\chi_{35539}(1284,\cdot)\) \(\chi_{35539}(1314,\cdot)\) \(\chi_{35539}(1333,\cdot)\) ...

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: $\Q(\zeta_{1269})$
Fixed field: Number field defined by a degree 2538 polynomial (not computed)

Values on generators

\((5078,15233)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1877}{2538}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 35539 }(887, a) \) \(-1\)\(1\)\(e\left(\frac{1031}{2538}\right)\)\(e\left(\frac{667}{846}\right)\)\(e\left(\frac{1031}{1269}\right)\)\(e\left(\frac{115}{141}\right)\)\(e\left(\frac{247}{1269}\right)\)\(e\left(\frac{185}{846}\right)\)\(e\left(\frac{244}{423}\right)\)\(e\left(\frac{563}{2538}\right)\)\(e\left(\frac{1201}{2538}\right)\)\(e\left(\frac{1525}{2538}\right)\)
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)
 
\( \chi_{ 35539 }(887,a) \;\) at \(\;a = \) e.g. 2