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Properties

Label 5400.3029
Modulus $5400$
Conductor $5400$
Order $90$
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(5400, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([0,45,25,9]))
 
Copy content gp:[g,chi] = znchar(Mod(3029, 5400))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5400.3029");
 

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: \(5400\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
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 5400.fc

\(\chi_{5400}(29,\cdot)\) \(\chi_{5400}(389,\cdot)\) \(\chi_{5400}(509,\cdot)\) \(\chi_{5400}(869,\cdot)\) \(\chi_{5400}(1109,\cdot)\) \(\chi_{5400}(1229,\cdot)\) \(\chi_{5400}(1469,\cdot)\) \(\chi_{5400}(1589,\cdot)\) \(\chi_{5400}(1829,\cdot)\) \(\chi_{5400}(2189,\cdot)\) \(\chi_{5400}(2309,\cdot)\) \(\chi_{5400}(2669,\cdot)\) \(\chi_{5400}(2909,\cdot)\) \(\chi_{5400}(3029,\cdot)\) \(\chi_{5400}(3269,\cdot)\) \(\chi_{5400}(3389,\cdot)\) \(\chi_{5400}(3629,\cdot)\) \(\chi_{5400}(3989,\cdot)\) \(\chi_{5400}(4109,\cdot)\) \(\chi_{5400}(4469,\cdot)\) \(\chi_{5400}(4709,\cdot)\) \(\chi_{5400}(4829,\cdot)\) \(\chi_{5400}(5069,\cdot)\) \(\chi_{5400}(5189,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((1351,2701,1001,2377)\) → \((1,-1,e\left(\frac{5}{18}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5400 }(3029, a) \) \(-1\)\(1\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{90}\right)\)
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)
 
\( \chi_{ 5400 }(3029,a) \;\) at \(\;a = \) e.g. 2