Properties

Label 50.13
Modulus $50$
Conductor $25$
Order $20$
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(50)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([19]))
 
pari: [g,chi] = znchar(Mod(13,50))
 

Basic properties

Modulus: \(50\)
Conductor: \(25\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{25}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 50.f

\(\chi_{50}(3,\cdot)\) \(\chi_{50}(13,\cdot)\) \(\chi_{50}(17,\cdot)\) \(\chi_{50}(23,\cdot)\) \(\chi_{50}(27,\cdot)\) \(\chi_{50}(33,\cdot)\) \(\chi_{50}(37,\cdot)\) \(\chi_{50}(47,\cdot)\)

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

\(27\) → \(e\left(\frac{19}{20}\right)\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(-1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(-i\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{19}{20}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{25})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 50 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{50}(13,\cdot)) = \sum_{r\in \Z/50\Z} \chi_{50}(13,r) e\left(\frac{r}{25}\right) = 4.9114362536+0.9369065729i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 50 }(13,·),\chi_{ 50 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{50}(13,\cdot),\chi_{50}(1,\cdot)) = \sum_{r\in \Z/50\Z} \chi_{50}(13,r) \chi_{50}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 50 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{50}(13,·)) = \sum_{r \in \Z/50\Z} \chi_{50}(13,r) e\left(\frac{1 r + 2 r^{-1}}{50}\right) = 0.7963162156+5.0277427122i \)