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Properties

Label 17661.1193
Modulus $17661$
Conductor $17661$
Order $1218$
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(17661, base_ring=CyclotomicField(1218)) M = H._module chi = DirichletCharacter(H, M([609,203,129]))
 
Copy content gp:[g,chi] = znchar(Mod(1193, 17661))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("17661.1193");
 

Basic properties

Modulus: \(17661\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(17661\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1218\)
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 17661.dh

\(\chi_{17661}(5,\cdot)\) \(\chi_{17661}(38,\cdot)\) \(\chi_{17661}(80,\cdot)\) \(\chi_{17661}(122,\cdot)\) \(\chi_{17661}(299,\cdot)\) \(\chi_{17661}(332,\cdot)\) \(\chi_{17661}(341,\cdot)\) \(\chi_{17661}(353,\cdot)\) \(\chi_{17661}(383,\cdot)\) \(\chi_{17661}(584,\cdot)\) \(\chi_{17661}(593,\cdot)\) \(\chi_{17661}(614,\cdot)\) \(\chi_{17661}(647,\cdot)\) \(\chi_{17661}(689,\cdot)\) \(\chi_{17661}(731,\cdot)\) \(\chi_{17661}(845,\cdot)\) \(\chi_{17661}(908,\cdot)\) \(\chi_{17661}(941,\cdot)\) \(\chi_{17661}(950,\cdot)\) \(\chi_{17661}(962,\cdot)\) \(\chi_{17661}(992,\cdot)\) \(\chi_{17661}(1193,\cdot)\) \(\chi_{17661}(1202,\cdot)\) \(\chi_{17661}(1223,\cdot)\) \(\chi_{17661}(1256,\cdot)\) \(\chi_{17661}(1298,\cdot)\) \(\chi_{17661}(1340,\cdot)\) \(\chi_{17661}(1454,\cdot)\) \(\chi_{17661}(1517,\cdot)\) \(\chi_{17661}(1550,\cdot)\)

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: $\Q(\zeta_{609})$
Fixed field: Number field defined by a degree 1218 polynomial (not computed)

Values on generators

\((5888,7570,16822)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{43}{406}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 17661 }(1193, a) \) \(1\)\(1\)\(e\left(\frac{572}{609}\right)\)\(e\left(\frac{535}{609}\right)\)\(e\left(\frac{194}{609}\right)\)\(e\left(\frac{166}{203}\right)\)\(e\left(\frac{157}{609}\right)\)\(e\left(\frac{454}{609}\right)\)\(e\left(\frac{249}{406}\right)\)\(e\left(\frac{461}{609}\right)\)\(e\left(\frac{59}{174}\right)\)\(e\left(\frac{17}{609}\right)\)
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)
 
\( \chi_{ 17661 }(1193,a) \;\) at \(\;a = \) e.g. 2