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Properties

Label 43681.26381
Modulus $43681$
Conductor $209$
Order $45$
Real no
Primitive no
Minimal no
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(43681, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([72,40]))
 
Copy content gp:[g,chi] = znchar(Mod(26381, 43681))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("43681.26381");
 

Basic properties

Modulus: \(43681\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(209\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(45\)
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 \(\chi_{209}(47,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 43681.bd

\(\chi_{43681}(245,\cdot)\) \(\chi_{43681}(3348,\cdot)\) \(\chi_{43681}(6010,\cdot)\) \(\chi_{43681}(12302,\cdot)\) \(\chi_{43681}(13095,\cdot)\) \(\chi_{43681}(15757,\cdot)\) \(\chi_{43681}(16344,\cdot)\) \(\chi_{43681}(17021,\cdot)\) \(\chi_{43681}(17427,\cdot)\) \(\chi_{43681}(18473,\cdot)\) \(\chi_{43681}(19006,\cdot)\) \(\chi_{43681}(20089,\cdot)\) \(\chi_{43681}(22049,\cdot)\) \(\chi_{43681}(25298,\cdot)\) \(\chi_{43681}(26381,\cdot)\) \(\chi_{43681}(26768,\cdot)\) \(\chi_{43681}(28220,\cdot)\) \(\chi_{43681}(29847,\cdot)\) \(\chi_{43681}(30017,\cdot)\) \(\chi_{43681}(31100,\cdot)\) \(\chi_{43681}(31469,\cdot)\) \(\chi_{43681}(32552,\cdot)\) \(\chi_{43681}(39594,\cdot)\) \(\chi_{43681}(42843,\cdot)\)

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: 45.45.43679806300610465846484971330073185012597520004657724600953543350870941304329239684756561.1

Values on generators

\((21661,22023)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{4}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 43681 }(26381, a) \) \(1\)\(1\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{3}\right)\)
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)
 
\( \chi_{ 43681 }(26381,a) \;\) at \(\;a = \) e.g. 2