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Properties

Label 47311.6668
Modulus $47311$
Conductor $47311$
Order $220$
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(47311, base_ring=CyclotomicField(220)) M = H._module chi = DirichletCharacter(H, M([202,165,130]))
 
Copy content gp:[g,chi] = znchar(Mod(6668, 47311))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("47311.6668");
 

Basic properties

Modulus: \(47311\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(47311\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(220\)
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 47311.on

\(\chi_{47311}(149,\cdot)\) \(\chi_{47311}(778,\cdot)\) \(\chi_{47311}(931,\cdot)\) \(\chi_{47311}(1602,\cdot)\) \(\chi_{47311}(1993,\cdot)\) \(\chi_{47311}(2274,\cdot)\) \(\chi_{47311}(4237,\cdot)\) \(\chi_{47311}(4858,\cdot)\) \(\chi_{47311}(5997,\cdot)\) \(\chi_{47311}(6243,\cdot)\) \(\chi_{47311}(6668,\cdot)\) \(\chi_{47311}(7850,\cdot)\) \(\chi_{47311}(7926,\cdot)\) \(\chi_{47311}(8071,\cdot)\) \(\chi_{47311}(8989,\cdot)\) \(\chi_{47311}(10094,\cdot)\) \(\chi_{47311}(10315,\cdot)\) \(\chi_{47311}(10808,\cdot)\) \(\chi_{47311}(11369,\cdot)\) \(\chi_{47311}(11998,\cdot)\) \(\chi_{47311}(12108,\cdot)\) \(\chi_{47311}(12542,\cdot)\) \(\chi_{47311}(12601,\cdot)\) \(\chi_{47311}(12729,\cdot)\) \(\chi_{47311}(12899,\cdot)\) \(\chi_{47311}(12933,\cdot)\) \(\chi_{47311}(13009,\cdot)\) \(\chi_{47311}(13196,\cdot)\) \(\chi_{47311}(13230,\cdot)\) \(\chi_{47311}(13723,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((5084,8350,10286)\) → \((e\left(\frac{101}{110}\right),-i,e\left(\frac{13}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 47311 }(6668, a) \) \(1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{220}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{63}{220}\right)\)\(e\left(\frac{133}{220}\right)\)\(e\left(\frac{199}{220}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{110}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{9}{44}\right)\)
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)
 
\( \chi_{ 47311 }(6668,a) \;\) at \(\;a = \) e.g. 2