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Properties

Label 1659.950
Modulus $1659$
Conductor $1659$
Order $78$
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(1659, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([39,65,4]))
 
Copy content gp:[g,chi] = znchar(Mod(950, 1659))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1659.950");
 

Basic properties

Modulus: \(1659\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1659\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(78\)
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 1659.ce

\(\chi_{1659}(194,\cdot)\) \(\chi_{1659}(248,\cdot)\) \(\chi_{1659}(257,\cdot)\) \(\chi_{1659}(269,\cdot)\) \(\chi_{1659}(341,\cdot)\) \(\chi_{1659}(446,\cdot)\) \(\chi_{1659}(479,\cdot)\) \(\chi_{1659}(500,\cdot)\) \(\chi_{1659}(584,\cdot)\) \(\chi_{1659}(626,\cdot)\) \(\chi_{1659}(677,\cdot)\) \(\chi_{1659}(794,\cdot)\) \(\chi_{1659}(803,\cdot)\) \(\chi_{1659}(866,\cdot)\) \(\chi_{1659}(878,\cdot)\) \(\chi_{1659}(941,\cdot)\) \(\chi_{1659}(950,\cdot)\) \(\chi_{1659}(992,\cdot)\) \(\chi_{1659}(1046,\cdot)\) \(\chi_{1659}(1067,\cdot)\) \(\chi_{1659}(1076,\cdot)\) \(\chi_{1659}(1235,\cdot)\) \(\chi_{1659}(1517,\cdot)\) \(\chi_{1659}(1622,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((554,1186,1030)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{2}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1659 }(950, a) \) \(1\)\(1\)\(e\left(\frac{29}{78}\right)\)\(e\left(\frac{29}{39}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{19}{78}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{21}{26}\right)\)
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)
 
\( \chi_{ 1659 }(950,a) \;\) at \(\;a = \) e.g. 2