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Properties

Label 17593.8202
Modulus $17593$
Conductor $17593$
Order $72$
Real no
Primitive yes
Minimal yes
Parity odd

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(17593, base_ring=CyclotomicField(72)) M = H._module chi = DirichletCharacter(H, M([67,27]))
 
Copy content gp:[g,chi] = znchar(Mod(8202, 17593))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("17593.8202");
 

Basic properties

Modulus: \(17593\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(17593\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(72\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 17593.ks

\(\chi_{17593}(2161,\cdot)\) \(\chi_{17593}(2643,\cdot)\) \(\chi_{17593}(2659,\cdot)\) \(\chi_{17593}(3344,\cdot)\) \(\chi_{17593}(4127,\cdot)\) \(\chi_{17593}(4790,\cdot)\) \(\chi_{17593}(5535,\cdot)\) \(\chi_{17593}(6477,\cdot)\) \(\chi_{17593}(6537,\cdot)\) \(\chi_{17593}(6997,\cdot)\) \(\chi_{17593}(7441,\cdot)\) \(\chi_{17593}(8202,\cdot)\) \(\chi_{17593}(9391,\cdot)\) \(\chi_{17593}(10152,\cdot)\) \(\chi_{17593}(10596,\cdot)\) \(\chi_{17593}(11056,\cdot)\) \(\chi_{17593}(11116,\cdot)\) \(\chi_{17593}(12058,\cdot)\) \(\chi_{17593}(12803,\cdot)\) \(\chi_{17593}(13466,\cdot)\) \(\chi_{17593}(14249,\cdot)\) \(\chi_{17593}(14934,\cdot)\) \(\chi_{17593}(14950,\cdot)\) \(\chi_{17593}(15432,\cdot)\)

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: $\Q(\zeta_{72})$
Fixed field: Number field defined by a degree 72 polynomial

Values on generators

\((9641,3140)\) → \((e\left(\frac{67}{72}\right),e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 17593 }(8202, a) \) \(-1\)\(1\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{49}{72}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{9}\right)\)
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)
 
\( \chi_{ 17593 }(8202,a) \;\) at \(\;a = \) e.g. 2