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Properties

Label 2700.1069
Modulus $2700$
Conductor $675$
Order $90$
Real no
Primitive no
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(2700, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([0,20,81]))
 
Copy content gp:[g,chi] = znchar(Mod(1069, 2700))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.1069");
 

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: \(675\)
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: no, induced from \(\chi_{675}(394,\cdot)\)
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: 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 2700.cl

\(\chi_{2700}(169,\cdot)\) \(\chi_{2700}(229,\cdot)\) \(\chi_{2700}(409,\cdot)\) \(\chi_{2700}(529,\cdot)\) \(\chi_{2700}(589,\cdot)\) \(\chi_{2700}(709,\cdot)\) \(\chi_{2700}(769,\cdot)\) \(\chi_{2700}(889,\cdot)\) \(\chi_{2700}(1069,\cdot)\) \(\chi_{2700}(1129,\cdot)\) \(\chi_{2700}(1309,\cdot)\) \(\chi_{2700}(1429,\cdot)\) \(\chi_{2700}(1489,\cdot)\) \(\chi_{2700}(1609,\cdot)\) \(\chi_{2700}(1669,\cdot)\) \(\chi_{2700}(1789,\cdot)\) \(\chi_{2700}(1969,\cdot)\) \(\chi_{2700}(2029,\cdot)\) \(\chi_{2700}(2209,\cdot)\) \(\chi_{2700}(2329,\cdot)\) \(\chi_{2700}(2389,\cdot)\) \(\chi_{2700}(2509,\cdot)\) \(\chi_{2700}(2569,\cdot)\) \(\chi_{2700}(2689,\cdot)\)

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: $\Q(\zeta_{45})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 90 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1351,1001,2377)\) → \((1,e\left(\frac{2}{9}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2700 }(1069, a) \) \(1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{29}{45}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{17}{45}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 2700 }(1069,a) \;\) at \(\;a = \) e.g. 2