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Properties

Label 32851.2453
Modulus $32851$
Conductor $32851$
Order $342$
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(32851, base_ring=CyclotomicField(342)) M = H._module chi = DirichletCharacter(H, M([57,228,217]))
 
Copy content gp:[g,chi] = znchar(Mod(2453, 32851))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32851.2453");
 

Basic properties

Modulus: \(32851\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(32851\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(342\)
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 32851.ro

\(\chi_{32851}(269,\cdot)\) \(\chi_{32851}(451,\cdot)\) \(\chi_{32851}(523,\cdot)\) \(\chi_{32851}(724,\cdot)\) \(\chi_{32851}(887,\cdot)\) \(\chi_{32851}(1998,\cdot)\) \(\chi_{32851}(2180,\cdot)\) \(\chi_{32851}(2252,\cdot)\) \(\chi_{32851}(2453,\cdot)\) \(\chi_{32851}(2616,\cdot)\) \(\chi_{32851}(3435,\cdot)\) \(\chi_{32851}(3727,\cdot)\) \(\chi_{32851}(3981,\cdot)\) \(\chi_{32851}(4182,\cdot)\) \(\chi_{32851}(4345,\cdot)\) \(\chi_{32851}(5164,\cdot)\) \(\chi_{32851}(5456,\cdot)\) \(\chi_{32851}(5638,\cdot)\) \(\chi_{32851}(5710,\cdot)\) \(\chi_{32851}(5911,\cdot)\) \(\chi_{32851}(6074,\cdot)\) \(\chi_{32851}(6893,\cdot)\) \(\chi_{32851}(7185,\cdot)\) \(\chi_{32851}(7367,\cdot)\) \(\chi_{32851}(7439,\cdot)\) \(\chi_{32851}(7640,\cdot)\) \(\chi_{32851}(7803,\cdot)\) \(\chi_{32851}(8622,\cdot)\) \(\chi_{32851}(8914,\cdot)\) \(\chi_{32851}(9096,\cdot)\) ...

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: $\Q(\zeta_{171})$
Fixed field: Number field defined by a degree 342 polynomial (not computed)

Values on generators

\((14080,20217,22023)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{3}\right),e\left(\frac{217}{342}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 32851 }(2453, a) \) \(1\)\(1\)\(e\left(\frac{217}{342}\right)\)\(e\left(\frac{5}{171}\right)\)\(e\left(\frac{46}{171}\right)\)\(e\left(\frac{13}{342}\right)\)\(e\left(\frac{227}{342}\right)\)\(e\left(\frac{103}{114}\right)\)\(e\left(\frac{10}{171}\right)\)\(e\left(\frac{115}{171}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{17}{57}\right)\)
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)
 
\( \chi_{ 32851 }(2453,a) \;\) at \(\;a = \) e.g. 2