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Properties

Label 2521.1897
Modulus $2521$
Conductor $2521$
Order $45$
Real no
Primitive yes
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(2521, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([88]))
 
Copy content gp:[g,chi] = znchar(Mod(1897, 2521))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2521.1897");
 

Basic properties

Modulus: \(2521\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2521\)
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: 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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2521.x

\(\chi_{2521}(101,\cdot)\) \(\chi_{2521}(108,\cdot)\) \(\chi_{2521}(117,\cdot)\) \(\chi_{2521}(189,\cdot)\) \(\chi_{2521}(270,\cdot)\) \(\chi_{2521}(427,\cdot)\) \(\chi_{2521}(610,\cdot)\) \(\chi_{2521}(738,\cdot)\) \(\chi_{2521}(807,\cdot)\) \(\chi_{2521}(817,\cdot)\) \(\chi_{2521}(824,\cdot)\) \(\chi_{2521}(827,\cdot)\) \(\chi_{2521}(831,\cdot)\) \(\chi_{2521}(1084,\cdot)\) \(\chi_{2521}(1142,\cdot)\) \(\chi_{2521}(1263,\cdot)\) \(\chi_{2521}(1513,\cdot)\) \(\chi_{2521}(1525,\cdot)\) \(\chi_{2521}(1580,\cdot)\) \(\chi_{2521}(1897,\cdot)\) \(\chi_{2521}(1945,\cdot)\) \(\chi_{2521}(1955,\cdot)\) \(\chi_{2521}(2312,\cdot)\) \(\chi_{2521}(2328,\cdot)\)

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 45 polynomial

Values on generators

\(17\) → \(e\left(\frac{44}{45}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2521 }(1897, a) \) \(1\)\(1\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{32}{45}\right)\)\(1\)
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)
 
\( \chi_{ 2521 }(1897,a) \;\) at \(\;a = \) e.g. 2