Properties

Label 25.18
Modulus $25$
Conductor $5$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(25)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3]))
 
pari: [g,chi] = znchar(Mod(18,25))
 

Basic properties

Modulus: \(25\)
Conductor: \(5\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{5}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 25.c

\(\chi_{25}(7,\cdot)\) \(\chi_{25}(18,\cdot)\)

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

Values on generators

\(2\) → \(-i\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(-i\)\(i\)\(-1\)\(1\)\(-i\)\(i\)\(-1\)\(1\)\(-i\)\(i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 25 }(18,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{25}(18,\cdot)) = \sum_{r\in \Z/25\Z} \chi_{25}(18,r) e\left(\frac{2r}{25}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 25 }(18,·),\chi_{ 25 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{25}(18,\cdot),\chi_{25}(1,\cdot)) = \sum_{r\in \Z/25\Z} \chi_{25}(18,r) \chi_{25}(1,1-r) = -5 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 25 }(18,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{25}(18,·)) = \sum_{r \in \Z/25\Z} \chi_{25}(18,r) e\left(\frac{1 r + 2 r^{-1}}{25}\right) = 0.0 \)