Properties

Modulus 75
Conductor 75
Order 20
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 75.l

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([10,11]))
 
pari: [g,chi] = znchar(Mod(23,75))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 75
Conductor = 75
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 20
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 = even
Orbit label = 75.l
Orbit index = 12

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 ]
 

\(\chi_{75}(2,\cdot)\) \(\chi_{75}(8,\cdot)\) \(\chi_{75}(17,\cdot)\) \(\chi_{75}(23,\cdot)\) \(\chi_{75}(38,\cdot)\) \(\chi_{75}(47,\cdot)\) \(\chi_{75}(53,\cdot)\) \(\chi_{75}(62,\cdot)\)

Values on generators

\((26,52)\) → \((-1,e\left(\frac{11}{20}\right))\)

Values

-112478111314161719
\(1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(-i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{20})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 75 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{75}(23,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(23,r) e\left(\frac{2r}{75}\right) = 8.5919653481+1.0854176421i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 75 }(23,·),\chi_{ 75 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{75}(23,\cdot),\chi_{75}(1,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(23,r) \chi_{75}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 75 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{75}(23,·)) = \sum_{r \in \Z/75\Z} \chi_{75}(23,r) e\left(\frac{1 r + 2 r^{-1}}{75}\right) = 0.0 \)