Properties

Modulus 104
Conductor 104
Order 4
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 104.j

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(104)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,2,3]))
 
pari: [g,chi] = znchar(Mod(5,104))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 104
Conductor = 104
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 4
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 104.j
Orbit index = 10

Galois orbit

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

\(\chi_{104}(5,\cdot)\) \(\chi_{104}(21,\cdot)\)

Values on generators

\((79,53,41)\) → \((1,-1,-i)\)

Values

-113579111517192123
\(-1\)\(1\)\(-1\)\(i\)\(i\)\(1\)\(-i\)\(-i\)\(-1\)\(i\)\(-i\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 104 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{104}(5,\cdot)) = \sum_{r\in \Z/104\Z} \chi_{104}(5,r) e\left(\frac{r}{52}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 104 }(5,·),\chi_{ 104 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{104}(5,\cdot),\chi_{104}(1,\cdot)) = \sum_{r\in \Z/104\Z} \chi_{104}(5,r) \chi_{104}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 104 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{104}(5,·)) = \sum_{r \in \Z/104\Z} \chi_{104}(5,r) e\left(\frac{1 r + 2 r^{-1}}{104}\right) = 1.6687616167+1.6687616167i \)