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Properties

Label 2205.638
Modulus $2205$
Conductor $15$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,3,0]))
 
Copy content pari:[g,chi] = znchar(Mod(638,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(15\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{15}(8,\cdot)\)
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.m

\(\chi_{2205}(197,\cdot)\) \(\chi_{2205}(638,\cdot)\)

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: \(\mathbb{Q}(i)\)
Fixed field: \(\Q(\zeta_{15})^+\)

Values on generators

\((1226,442,1081)\) → \((-1,-i,1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 2205 }(638, a) \) \(1\)\(1\)\(i\)\(-1\)\(-i\)\(-1\)\(i\)\(1\)\(i\)\(-1\)\(-i\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2205 }(638,a) \;\) at \(\;a = \) e.g. 2