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Properties

Label 5400.47
Modulus $5400$
Conductor $2700$
Order $180$
Real no
Primitive no
Minimal no
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(5400, base_ring=CyclotomicField(180)) M = H._module chi = DirichletCharacter(H, M([90,0,70,153]))
 
Copy content gp:[g,chi] = znchar(Mod(47, 5400))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5400.47");
 

Basic properties

Modulus: \(5400\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2700\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(180\)
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 \(\chi_{2700}(47,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 5400.fn

\(\chi_{5400}(23,\cdot)\) \(\chi_{5400}(47,\cdot)\) \(\chi_{5400}(167,\cdot)\) \(\chi_{5400}(263,\cdot)\) \(\chi_{5400}(383,\cdot)\) \(\chi_{5400}(527,\cdot)\) \(\chi_{5400}(623,\cdot)\) \(\chi_{5400}(767,\cdot)\) \(\chi_{5400}(887,\cdot)\) \(\chi_{5400}(983,\cdot)\) \(\chi_{5400}(1103,\cdot)\) \(\chi_{5400}(1127,\cdot)\) \(\chi_{5400}(1247,\cdot)\) \(\chi_{5400}(1463,\cdot)\) \(\chi_{5400}(1487,\cdot)\) \(\chi_{5400}(1703,\cdot)\) \(\chi_{5400}(1823,\cdot)\) \(\chi_{5400}(1847,\cdot)\) \(\chi_{5400}(1967,\cdot)\) \(\chi_{5400}(2063,\cdot)\) \(\chi_{5400}(2183,\cdot)\) \(\chi_{5400}(2327,\cdot)\) \(\chi_{5400}(2423,\cdot)\) \(\chi_{5400}(2567,\cdot)\) \(\chi_{5400}(2687,\cdot)\) \(\chi_{5400}(2783,\cdot)\) \(\chi_{5400}(2903,\cdot)\) \(\chi_{5400}(2927,\cdot)\) \(\chi_{5400}(3047,\cdot)\) \(\chi_{5400}(3263,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((1351,2701,1001,2377)\) → \((-1,1,e\left(\frac{7}{18}\right),e\left(\frac{17}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5400 }(47, a) \) \(-1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{47}{180}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{23}{180}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{7}{90}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{1}{90}\right)\)
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)
 
\( \chi_{ 5400 }(47,a) \;\) at \(\;a = \) e.g. 2