Properties

Modulus 75
Conductor 25
Order 10
Real no
Primitive no
Minimal yes
Parity even
Orbit label 75.i

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(75)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9]))
 
pari: [g,chi] = znchar(Mod(19,75))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 75
Conductor = 25
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 10
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 75.i
Orbit index = 9

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 ]
 

\(\chi_{75}(4,\cdot)\) \(\chi_{75}(19,\cdot)\) \(\chi_{75}(34,\cdot)\) \(\chi_{75}(64,\cdot)\)

Values on generators

\((26,52)\) → \((1,e\left(\frac{9}{10}\right))\)

Values

-112478111314161719
\(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(-1\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{5}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 75 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{75}(19,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(19,r) e\left(\frac{2r}{75}\right) = 3.1871199487+-3.8525662139i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 75 }(19,·),\chi_{ 75 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{75}(19,\cdot),\chi_{75}(1,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(19,r) \chi_{75}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 75 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{75}(19,·)) = \sum_{r \in \Z/75\Z} \chi_{75}(19,r) e\left(\frac{1 r + 2 r^{-1}}{75}\right) = 0.0 \)