↑

Properties

Label 2809.40
Modulus $2809$
Conductor $2809$
Order $1378$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2809, base_ring=CyclotomicField(1378)) M = H._module chi = DirichletCharacter(H, M([805]))
 
Copy content gp:[g,chi] = znchar(Mod(40, 2809))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2809.40");
 

Basic properties

Modulus: \(2809\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2809\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1378\)
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 2809.k

\(\chi_{2809}(4,\cdot)\) \(\chi_{2809}(6,\cdot)\) \(\chi_{2809}(7,\cdot)\) \(\chi_{2809}(9,\cdot)\) \(\chi_{2809}(11,\cdot)\) \(\chi_{2809}(17,\cdot)\) \(\chi_{2809}(25,\cdot)\) \(\chi_{2809}(29,\cdot)\) \(\chi_{2809}(37,\cdot)\) \(\chi_{2809}(38,\cdot)\) \(\chi_{2809}(40,\cdot)\) \(\chi_{2809}(43,\cdot)\) \(\chi_{2809}(57,\cdot)\) \(\chi_{2809}(59,\cdot)\) \(\chi_{2809}(60,\cdot)\) \(\chi_{2809}(62,\cdot)\) \(\chi_{2809}(64,\cdot)\) \(\chi_{2809}(70,\cdot)\) \(\chi_{2809}(78,\cdot)\) \(\chi_{2809}(82,\cdot)\) \(\chi_{2809}(90,\cdot)\) \(\chi_{2809}(91,\cdot)\) \(\chi_{2809}(93,\cdot)\) \(\chi_{2809}(96,\cdot)\) \(\chi_{2809}(110,\cdot)\) \(\chi_{2809}(112,\cdot)\) \(\chi_{2809}(113,\cdot)\) \(\chi_{2809}(115,\cdot)\) \(\chi_{2809}(117,\cdot)\) \(\chi_{2809}(123,\cdot)\) ...

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: $\Q(\zeta_{689})$
Fixed field: Number field defined by a degree 1378 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{805}{1378}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2809 }(40, a) \) \(1\)\(1\)\(e\left(\frac{805}{1378}\right)\)\(e\left(\frac{451}{1378}\right)\)\(e\left(\frac{116}{689}\right)\)\(e\left(\frac{1071}{1378}\right)\)\(e\left(\frac{628}{689}\right)\)\(e\left(\frac{526}{689}\right)\)\(e\left(\frac{1037}{1378}\right)\)\(e\left(\frac{451}{689}\right)\)\(e\left(\frac{249}{689}\right)\)\(e\left(\frac{62}{689}\right)\)
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)
 
\( \chi_{ 2809 }(40,a) \;\) at \(\;a = \) e.g. 2