↑

Properties

Label 1592.27
Modulus $1592$
Conductor $1592$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1592, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([33,33,1]))
 
Copy content gp:[g,chi] = znchar(Mod(27, 1592))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1592.27");
 

Basic properties

Modulus: \(1592\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1592\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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 1592.bj

\(\chi_{1592}(27,\cdot)\) \(\chi_{1592}(59,\cdot)\) \(\chi_{1592}(67,\cdot)\) \(\chi_{1592}(83,\cdot)\) \(\chi_{1592}(147,\cdot)\) \(\chi_{1592}(171,\cdot)\) \(\chi_{1592}(211,\cdot)\) \(\chi_{1592}(275,\cdot)\) \(\chi_{1592}(499,\cdot)\) \(\chi_{1592}(507,\cdot)\) \(\chi_{1592}(771,\cdot)\) \(\chi_{1592}(851,\cdot)\) \(\chi_{1592}(931,\cdot)\) \(\chi_{1592}(955,\cdot)\) \(\chi_{1592}(987,\cdot)\) \(\chi_{1592}(1083,\cdot)\) \(\chi_{1592}(1211,\cdot)\) \(\chi_{1592}(1435,\cdot)\) \(\chi_{1592}(1475,\cdot)\) \(\chi_{1592}(1587,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((399,797,401)\) → \((-1,-1,e\left(\frac{1}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1592 }(27, a) \) \(1\)\(1\)\(e\left(\frac{1}{66}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{7}{66}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)
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)
 
\( \chi_{ 1592 }(27,a) \;\) at \(\;a = \) e.g. 2