Properties

Label 572.135
Modulus $572$
Conductor $572$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(572, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,16,15]))
 
pari: [g,chi] = znchar(Mod(135,572))
 

Basic properties

Modulus: \(572\)
Conductor: \(572\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 572.bk

\(\chi_{572}(31,\cdot)\) \(\chi_{572}(47,\cdot)\) \(\chi_{572}(135,\cdot)\) \(\chi_{572}(203,\cdot)\) \(\chi_{572}(291,\cdot)\) \(\chi_{572}(411,\cdot)\) \(\chi_{572}(499,\cdot)\) \(\chi_{572}(515,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: 20.20.2466227638053818291661582077909665841152.1

Values on generators

\((287,365,353)\) → \((-1,e\left(\frac{4}{5}\right),-i)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(15\)\(17\)\(19\)\(21\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(i\)\(1\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 572 }(135,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 572 }(135,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 572 }(135,·),\chi_{ 572 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 572 }(135,·)) \;\) at \(\; a,b = \) e.g. 1,2