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Properties

Label 4009.3092
Modulus $4009$
Conductor $4009$
Order $126$
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(4009, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([49,39]))
 
Copy content gp:[g,chi] = znchar(Mod(3092, 4009))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4009.3092");
 

Basic properties

Modulus: \(4009\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4009\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(126\)
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 4009.dz

\(\chi_{4009}(110,\cdot)\) \(\chi_{4009}(249,\cdot)\) \(\chi_{4009}(516,\cdot)\) \(\chi_{4009}(659,\cdot)\) \(\chi_{4009}(801,\cdot)\) \(\chi_{4009}(870,\cdot)\) \(\chi_{4009}(876,\cdot)\) \(\chi_{4009}(1001,\cdot)\) \(\chi_{4009}(1021,\cdot)\) \(\chi_{4009}(1212,\cdot)\) \(\chi_{4009}(1515,\cdot)\) \(\chi_{4009}(1587,\cdot)\) \(\chi_{4009}(1826,\cdot)\) \(\chi_{4009}(1865,\cdot)\) \(\chi_{4009}(1932,\cdot)\) \(\chi_{4009}(2141,\cdot)\) \(\chi_{4009}(2142,\cdot)\) \(\chi_{4009}(2143,\cdot)\) \(\chi_{4009}(2160,\cdot)\) \(\chi_{4009}(2347,\cdot)\) \(\chi_{4009}(2352,\cdot)\) \(\chi_{4009}(2359,\cdot)\) \(\chi_{4009}(2371,\cdot)\) \(\chi_{4009}(2415,\cdot)\) \(\chi_{4009}(2689,\cdot)\) \(\chi_{4009}(2700,\cdot)\) \(\chi_{4009}(2986,\cdot)\) \(\chi_{4009}(3092,\cdot)\) \(\chi_{4009}(3620,\cdot)\) \(\chi_{4009}(3681,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((2111,1901)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{13}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4009 }(3092, a) \) \(1\)\(1\)\(e\left(\frac{44}{63}\right)\)\(e\left(\frac{23}{63}\right)\)\(e\left(\frac{25}{63}\right)\)\(e\left(\frac{5}{63}\right)\)\(e\left(\frac{4}{63}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{46}{63}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{17}{21}\right)\)
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)
 
\( \chi_{ 4009 }(3092,a) \;\) at \(\;a = \) e.g. 2