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Properties

Label 17161.9433
Modulus $17161$
Conductor $17161$
Order $131$
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(17161, base_ring=CyclotomicField(262)) M = H._module chi = DirichletCharacter(H, M([178]))
 
Copy content gp:[g,chi] = znchar(Mod(9433, 17161))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("17161.9433");
 

Basic properties

Modulus: \(17161\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(17161\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(131\)
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 17161.i

\(\chi_{17161}(132,\cdot)\) \(\chi_{17161}(263,\cdot)\) \(\chi_{17161}(394,\cdot)\) \(\chi_{17161}(525,\cdot)\) \(\chi_{17161}(656,\cdot)\) \(\chi_{17161}(787,\cdot)\) \(\chi_{17161}(918,\cdot)\) \(\chi_{17161}(1049,\cdot)\) \(\chi_{17161}(1180,\cdot)\) \(\chi_{17161}(1311,\cdot)\) \(\chi_{17161}(1442,\cdot)\) \(\chi_{17161}(1573,\cdot)\) \(\chi_{17161}(1704,\cdot)\) \(\chi_{17161}(1835,\cdot)\) \(\chi_{17161}(1966,\cdot)\) \(\chi_{17161}(2097,\cdot)\) \(\chi_{17161}(2228,\cdot)\) \(\chi_{17161}(2359,\cdot)\) \(\chi_{17161}(2490,\cdot)\) \(\chi_{17161}(2621,\cdot)\) \(\chi_{17161}(2752,\cdot)\) \(\chi_{17161}(2883,\cdot)\) \(\chi_{17161}(3014,\cdot)\) \(\chi_{17161}(3145,\cdot)\) \(\chi_{17161}(3276,\cdot)\) \(\chi_{17161}(3407,\cdot)\) \(\chi_{17161}(3538,\cdot)\) \(\chi_{17161}(3669,\cdot)\) \(\chi_{17161}(3800,\cdot)\) \(\chi_{17161}(3931,\cdot)\) ...

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: $\Q(\zeta_{131})$
Fixed field: Number field defined by a degree 131 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{89}{131}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 17161 }(9433, a) \) \(1\)\(1\)\(e\left(\frac{89}{131}\right)\)\(e\left(\frac{69}{131}\right)\)\(e\left(\frac{47}{131}\right)\)\(e\left(\frac{14}{131}\right)\)\(e\left(\frac{27}{131}\right)\)\(e\left(\frac{67}{131}\right)\)\(e\left(\frac{5}{131}\right)\)\(e\left(\frac{7}{131}\right)\)\(e\left(\frac{103}{131}\right)\)\(e\left(\frac{56}{131}\right)\)
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)
 
\( \chi_{ 17161 }(9433,a) \;\) at \(\;a = \) e.g. 2