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Properties

Label 39984.5189
Modulus $39984$
Conductor $39984$
Order $84$
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(39984, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([0,21,42,16,63]))
 
Copy content gp:[g,chi] = znchar(Mod(5189, 39984))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("39984.5189");
 

Basic properties

Modulus: \(39984\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(39984\)
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: 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 39984.qv

\(\chi_{39984}(2741,\cdot)\) \(\chi_{39984}(3005,\cdot)\) \(\chi_{39984}(5189,\cdot)\) \(\chi_{39984}(6269,\cdot)\) \(\chi_{39984}(8453,\cdot)\) \(\chi_{39984}(8717,\cdot)\) \(\chi_{39984}(10901,\cdot)\) \(\chi_{39984}(11981,\cdot)\) \(\chi_{39984}(14165,\cdot)\) \(\chi_{39984}(14429,\cdot)\) \(\chi_{39984}(16613,\cdot)\) \(\chi_{39984}(17693,\cdot)\) \(\chi_{39984}(19877,\cdot)\) \(\chi_{39984}(20141,\cdot)\) \(\chi_{39984}(23405,\cdot)\) \(\chi_{39984}(25589,\cdot)\) \(\chi_{39984}(28037,\cdot)\) \(\chi_{39984}(29117,\cdot)\) \(\chi_{39984}(31301,\cdot)\) \(\chi_{39984}(31565,\cdot)\) \(\chi_{39984}(33749,\cdot)\) \(\chi_{39984}(34829,\cdot)\) \(\chi_{39984}(37277,\cdot)\) \(\chi_{39984}(39461,\cdot)\)

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

\((24991,29989,26657,37537,14113)\) → \((1,i,-1,e\left(\frac{4}{21}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 39984 }(5189, a) \) \(-1\)\(1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{41}{84}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{3}{28}\right)\)
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)
 
\( \chi_{ 39984 }(5189,a) \;\) at \(\;a = \) e.g. 2