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Properties

Label 12221.3592
Modulus $12221$
Conductor $12221$
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(12221, base_ring=CyclotomicField(220)) M = H._module chi = DirichletCharacter(H, M([58,143]))
 
Copy content gp:[g,chi] = znchar(Mod(3592, 12221))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12221.3592");
 

Basic properties

Modulus: \(12221\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12221\)
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 12221.ef

\(\chi_{12221}(259,\cdot)\) \(\chi_{12221}(347,\cdot)\) \(\chi_{12221}(365,\cdot)\) \(\chi_{12221}(546,\cdot)\) \(\chi_{12221}(877,\cdot)\) \(\chi_{12221}(948,\cdot)\) \(\chi_{12221}(1042,\cdot)\) \(\chi_{12221}(1272,\cdot)\) \(\chi_{12221}(1370,\cdot)\) \(\chi_{12221}(1458,\cdot)\) \(\chi_{12221}(1476,\cdot)\) \(\chi_{12221}(1657,\cdot)\) \(\chi_{12221}(1988,\cdot)\) \(\chi_{12221}(2059,\cdot)\) \(\chi_{12221}(2153,\cdot)\) \(\chi_{12221}(2383,\cdot)\) \(\chi_{12221}(2481,\cdot)\) \(\chi_{12221}(2569,\cdot)\) \(\chi_{12221}(2587,\cdot)\) \(\chi_{12221}(2768,\cdot)\) \(\chi_{12221}(3099,\cdot)\) \(\chi_{12221}(3170,\cdot)\) \(\chi_{12221}(3494,\cdot)\) \(\chi_{12221}(3592,\cdot)\) \(\chi_{12221}(3680,\cdot)\) \(\chi_{12221}(3698,\cdot)\) \(\chi_{12221}(3879,\cdot)\) \(\chi_{12221}(4210,\cdot)\) \(\chi_{12221}(4281,\cdot)\) \(\chi_{12221}(4375,\cdot)\) ...

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

\((607,11617)\) → \((e\left(\frac{29}{110}\right),e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 12221 }(3592, a) \) \(1\)\(1\)\(e\left(\frac{201}{220}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{91}{110}\right)\)\(e\left(\frac{6}{55}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{153}{220}\right)\)\(e\left(\frac{163}{220}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{193}{220}\right)\)
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)
 
\( \chi_{ 12221 }(3592,a) \;\) at \(\;a = \) e.g. 2