↑

Properties

Label 2825.1081
Modulus $2825$
Conductor $2825$
Order $70$
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(2825, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([28,45]))
 
Copy content gp:[g,chi] = znchar(Mod(1081, 2825))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2825.1081");
 

Basic properties

Modulus: \(2825\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2825\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(70\)
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 2825.bx

\(\chi_{2825}(196,\cdot)\) \(\chi_{2825}(311,\cdot)\) \(\chi_{2825}(346,\cdot)\) \(\chi_{2825}(436,\cdot)\) \(\chi_{2825}(456,\cdot)\) \(\chi_{2825}(516,\cdot)\) \(\chi_{2825}(761,\cdot)\) \(\chi_{2825}(911,\cdot)\) \(\chi_{2825}(1021,\cdot)\) \(\chi_{2825}(1081,\cdot)\) \(\chi_{2825}(1441,\cdot)\) \(\chi_{2825}(1566,\cdot)\) \(\chi_{2825}(1586,\cdot)\) \(\chi_{2825}(1646,\cdot)\) \(\chi_{2825}(1891,\cdot)\) \(\chi_{2825}(2006,\cdot)\) \(\chi_{2825}(2041,\cdot)\) \(\chi_{2825}(2131,\cdot)\) \(\chi_{2825}(2211,\cdot)\) \(\chi_{2825}(2456,\cdot)\) \(\chi_{2825}(2571,\cdot)\) \(\chi_{2825}(2606,\cdot)\) \(\chi_{2825}(2696,\cdot)\) \(\chi_{2825}(2716,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((227,2376)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{9}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2825 }(1081, a) \) \(1\)\(1\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{31}{70}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{47}{70}\right)\)\(e\left(\frac{26}{35}\right)\)
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)
 
\( \chi_{ 2825 }(1081,a) \;\) at \(\;a = \) e.g. 2