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Properties

Label 17827.2
Modulus $17827$
Conductor $17827$
Order $17826$
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(17827, base_ring=CyclotomicField(17826)) M = H._module chi = DirichletCharacter(H, M([1]))
 
Copy content gp:[g,chi] = znchar(Mod(2, 17827))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("17827.2");
 

Basic properties

Modulus: \(17827\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(17827\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(17826\)
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 17827.h

\(\chi_{17827}(2,\cdot)\) \(\chi_{17827}(3,\cdot)\) \(\chi_{17827}(5,\cdot)\) \(\chi_{17827}(12,\cdot)\) \(\chi_{17827}(14,\cdot)\) \(\chi_{17827}(17,\cdot)\) \(\chi_{17827}(18,\cdot)\) \(\chi_{17827}(21,\cdot)\) \(\chi_{17827}(22,\cdot)\) \(\chi_{17827}(30,\cdot)\) \(\chi_{17827}(31,\cdot)\) \(\chi_{17827}(32,\cdot)\) \(\chi_{17827}(35,\cdot)\) \(\chi_{17827}(39,\cdot)\) \(\chi_{17827}(45,\cdot)\) \(\chi_{17827}(55,\cdot)\) \(\chi_{17827}(72,\cdot)\) \(\chi_{17827}(74,\cdot)\) \(\chi_{17827}(75,\cdot)\) \(\chi_{17827}(76,\cdot)\) \(\chi_{17827}(79,\cdot)\) \(\chi_{17827}(80,\cdot)\) \(\chi_{17827}(82,\cdot)\) \(\chi_{17827}(83,\cdot)\) \(\chi_{17827}(84,\cdot)\) \(\chi_{17827}(88,\cdot)\) \(\chi_{17827}(89,\cdot)\) \(\chi_{17827}(92,\cdot)\) \(\chi_{17827}(97,\cdot)\) \(\chi_{17827}(98,\cdot)\) ...

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: $\Q(\zeta_{8913})$
Fixed field: Number field defined by a degree 17826 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1}{17826}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 17827 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{1}{17826}\right)\)\(e\left(\frac{8369}{17826}\right)\)\(e\left(\frac{1}{8913}\right)\)\(e\left(\frac{10777}{17826}\right)\)\(e\left(\frac{1395}{2971}\right)\)\(e\left(\frac{446}{2971}\right)\)\(e\left(\frac{1}{5942}\right)\)\(e\left(\frac{8369}{8913}\right)\)\(e\left(\frac{5389}{8913}\right)\)\(e\left(\frac{1493}{8913}\right)\)
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)
 
\( \chi_{ 17827 }(2,a) \;\) at \(\;a = \) e.g. 2