↑

Properties

Label 1967.888
Modulus $1967$
Conductor $1967$
Order $70$
Real no
Primitive yes
Minimal yes
Parity odd

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(1967, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([35,47]))
 
Copy content gp:[g,chi] = znchar(Mod(888, 1967))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1967.888");
 

Basic properties

Modulus: \(1967\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1967\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1967.bt

\(\chi_{1967}(118,\cdot)\) \(\chi_{1967}(223,\cdot)\) \(\chi_{1967}(265,\cdot)\) \(\chi_{1967}(461,\cdot)\) \(\chi_{1967}(587,\cdot)\) \(\chi_{1967}(643,\cdot)\) \(\chi_{1967}(720,\cdot)\) \(\chi_{1967}(839,\cdot)\) \(\chi_{1967}(888,\cdot)\) \(\chi_{1967}(1126,\cdot)\) \(\chi_{1967}(1294,\cdot)\) \(\chi_{1967}(1413,\cdot)\) \(\chi_{1967}(1434,\cdot)\) \(\chi_{1967}(1448,\cdot)\) \(\chi_{1967}(1546,\cdot)\) \(\chi_{1967}(1588,\cdot)\) \(\chi_{1967}(1623,\cdot)\) \(\chi_{1967}(1651,\cdot)\) \(\chi_{1967}(1756,\cdot)\) \(\chi_{1967}(1805,\cdot)\) \(\chi_{1967}(1812,\cdot)\) \(\chi_{1967}(1882,\cdot)\) \(\chi_{1967}(1903,\cdot)\) \(\chi_{1967}(1917,\cdot)\)

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

\((563,1408)\) → \((-1,e\left(\frac{47}{70}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1967 }(888, a) \) \(-1\)\(1\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{27}{70}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{4}{35}\right)\)
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)
 
\( \chi_{ 1967 }(888,a) \;\) at \(\;a = \) e.g. 2