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Properties

Label 18288.7457
Modulus $18288$
Conductor $1143$
Order $126$
Real no
Primitive no
Minimal no
Parity even

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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(18288, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([0,0,105,83]))
 
Copy content gp:[g,chi] = znchar(Mod(7457, 18288))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("18288.7457");
 

Basic properties

Modulus: \(18288\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1143\)
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: no, induced from \(\chi_{1143}(599,\cdot)\)
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 18288.ne

\(\chi_{18288}(65,\cdot)\) \(\chi_{18288}(641,\cdot)\) \(\chi_{18288}(785,\cdot)\) \(\chi_{18288}(1073,\cdot)\) \(\chi_{18288}(1553,\cdot)\) \(\chi_{18288}(2801,\cdot)\) \(\chi_{18288}(2849,\cdot)\) \(\chi_{18288}(3281,\cdot)\) \(\chi_{18288}(3521,\cdot)\) \(\chi_{18288}(4385,\cdot)\) \(\chi_{18288}(4673,\cdot)\) \(\chi_{18288}(4865,\cdot)\) \(\chi_{18288}(6593,\cdot)\) \(\chi_{18288}(7457,\cdot)\) \(\chi_{18288}(8465,\cdot)\) \(\chi_{18288}(9329,\cdot)\) \(\chi_{18288}(9761,\cdot)\) \(\chi_{18288}(10145,\cdot)\) \(\chi_{18288}(10913,\cdot)\) \(\chi_{18288}(11777,\cdot)\) \(\chi_{18288}(11921,\cdot)\) \(\chi_{18288}(12161,\cdot)\) \(\chi_{18288}(12449,\cdot)\) \(\chi_{18288}(12785,\cdot)\) \(\chi_{18288}(13505,\cdot)\) \(\chi_{18288}(13601,\cdot)\) \(\chi_{18288}(13889,\cdot)\) \(\chi_{18288}(14321,\cdot)\) \(\chi_{18288}(14465,\cdot)\) \(\chi_{18288}(14945,\cdot)\) ...

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

\((2287,13717,8129,4321)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{83}{126}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 18288 }(7457, a) \) \(1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{11}{126}\right)\)\(e\left(\frac{79}{126}\right)\)\(e\left(\frac{37}{63}\right)\)\(e\left(\frac{67}{126}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{55}{63}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{17}{63}\right)\)\(e\left(\frac{61}{63}\right)\)
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)
 
\( \chi_{ 18288 }(7457,a) \;\) at \(\;a = \) e.g. 2