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Properties

Label 3339.2432
Modulus $3339$
Conductor $3339$
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(3339, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([13,13,66]))
 
Copy content gp:[g,chi] = znchar(Mod(2432, 3339))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3339.2432");
 

Basic properties

Modulus: \(3339\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3339\)
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 3339.dy

\(\chi_{3339}(68,\cdot)\) \(\chi_{3339}(227,\cdot)\) \(\chi_{3339}(572,\cdot)\) \(\chi_{3339}(731,\cdot)\) \(\chi_{3339}(950,\cdot)\) \(\chi_{3339}(1076,\cdot)\) \(\chi_{3339}(1109,\cdot)\) \(\chi_{3339}(1202,\cdot)\) \(\chi_{3339}(1235,\cdot)\) \(\chi_{3339}(1265,\cdot)\) \(\chi_{3339}(1361,\cdot)\) \(\chi_{3339}(1391,\cdot)\) \(\chi_{3339}(1424,\cdot)\) \(\chi_{3339}(1550,\cdot)\) \(\chi_{3339}(1706,\cdot)\) \(\chi_{3339}(1865,\cdot)\) \(\chi_{3339}(2273,\cdot)\) \(\chi_{3339}(2432,\cdot)\) \(\chi_{3339}(2462,\cdot)\) \(\chi_{3339}(2588,\cdot)\) \(\chi_{3339}(2621,\cdot)\) \(\chi_{3339}(2747,\cdot)\) \(\chi_{3339}(3155,\cdot)\) \(\chi_{3339}(3314,\cdot)\)

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

\((1856,955,1009)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{6}\right),e\left(\frac{11}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3339 }(2432, a) \) \(1\)\(1\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{71}{78}\right)\)\(e\left(\frac{11}{78}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{11}{78}\right)\)
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)
 
\( \chi_{ 3339 }(2432,a) \;\) at \(\;a = \) e.g. 2