↑

Properties

Label 2205.2204
Modulus $2205$
Conductor $105$
Order $2$
Real yes
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([1,1,1]))
 
Copy content pari:[g,chi] = znchar(Mod(2204,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(105\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(2\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{105}(104,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 2205.g

\(\chi_{2205}(2204,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{105}) \)

Values on generators

\((1226,442,1081)\) → \((-1,-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 2205 }(2204, a) \) \(1\)\(1\)\(1\)\(1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2205 }(2204,a) \;\) at \(\;a = \) e.g. 2