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Properties

Label 3920.1367
Modulus $3920$
Conductor $1960$
Order $84$
Real no
Primitive no
Minimal no
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(3920, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([42,42,21,16]))
 
Copy content gp:[g,chi] = znchar(Mod(1367, 3920))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3920.1367");
 

Basic properties

Modulus: \(3920\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1960\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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: no, induced from \(\chi_{1960}(387,\cdot)\)
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: no
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 3920.fy

\(\chi_{3920}(23,\cdot)\) \(\chi_{3920}(247,\cdot)\) \(\chi_{3920}(487,\cdot)\) \(\chi_{3920}(583,\cdot)\) \(\chi_{3920}(807,\cdot)\) \(\chi_{3920}(823,\cdot)\) \(\chi_{3920}(1143,\cdot)\) \(\chi_{3920}(1367,\cdot)\) \(\chi_{3920}(1383,\cdot)\) \(\chi_{3920}(1607,\cdot)\) \(\chi_{3920}(1703,\cdot)\) \(\chi_{3920}(1927,\cdot)\) \(\chi_{3920}(1943,\cdot)\) \(\chi_{3920}(2167,\cdot)\) \(\chi_{3920}(2263,\cdot)\) \(\chi_{3920}(2487,\cdot)\) \(\chi_{3920}(2503,\cdot)\) \(\chi_{3920}(2727,\cdot)\) \(\chi_{3920}(3047,\cdot)\) \(\chi_{3920}(3063,\cdot)\) \(\chi_{3920}(3287,\cdot)\) \(\chi_{3920}(3383,\cdot)\) \(\chi_{3920}(3623,\cdot)\) \(\chi_{3920}(3847,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1471,981,3137,3041)\) → \((-1,-1,i,e\left(\frac{4}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 3920 }(1367, a) \) \(1\)\(1\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{41}{84}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{5}{6}\right)\)
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)
 
\( \chi_{ 3920 }(1367,a) \;\) at \(\;a = \) e.g. 2