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Properties

Label 2700.439
Modulus $2700$
Conductor $2700$
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(2700, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([45,80,27]))
 
Copy content gp:[g,chi] = znchar(Mod(439, 2700))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.439");
 

Basic properties

Modulus: \(2700\)
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: \(90\)
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 2700.cj

\(\chi_{2700}(79,\cdot)\) \(\chi_{2700}(139,\cdot)\) \(\chi_{2700}(259,\cdot)\) \(\chi_{2700}(319,\cdot)\) \(\chi_{2700}(439,\cdot)\) \(\chi_{2700}(619,\cdot)\) \(\chi_{2700}(679,\cdot)\) \(\chi_{2700}(859,\cdot)\) \(\chi_{2700}(979,\cdot)\) \(\chi_{2700}(1039,\cdot)\) \(\chi_{2700}(1159,\cdot)\) \(\chi_{2700}(1219,\cdot)\) \(\chi_{2700}(1339,\cdot)\) \(\chi_{2700}(1519,\cdot)\) \(\chi_{2700}(1579,\cdot)\) \(\chi_{2700}(1759,\cdot)\) \(\chi_{2700}(1879,\cdot)\) \(\chi_{2700}(1939,\cdot)\) \(\chi_{2700}(2059,\cdot)\) \(\chi_{2700}(2119,\cdot)\) \(\chi_{2700}(2239,\cdot)\) \(\chi_{2700}(2419,\cdot)\) \(\chi_{2700}(2479,\cdot)\) \(\chi_{2700}(2659,\cdot)\)

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

\((1351,1001,2377)\) → \((-1,e\left(\frac{8}{9}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2700 }(439, a) \) \(-1\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{77}{90}\right)\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{14}{45}\right)\)
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)
 
\( \chi_{ 2700 }(439,a) \;\) at \(\;a = \) e.g. 2