Properties

Label 320.99
Modulus $320$
Conductor $320$
Order $16$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(320, base_ring=CyclotomicField(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8,11,8]))
 
pari: [g,chi] = znchar(Mod(99,320))
 

Basic properties

Modulus: \(320\)
Conductor: \(320\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 320.bh

\(\chi_{320}(19,\cdot)\) \(\chi_{320}(59,\cdot)\) \(\chi_{320}(99,\cdot)\) \(\chi_{320}(139,\cdot)\) \(\chi_{320}(179,\cdot)\) \(\chi_{320}(219,\cdot)\) \(\chi_{320}(259,\cdot)\) \(\chi_{320}(299,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.0.236118324143482260684800000000.1

Values on generators

\((191,261,257)\) → \((-1,e\left(\frac{11}{16}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(-1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(-i\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 320 }(99,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 320 }(99,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 320 }(99,·),\chi_{ 320 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 320 }(99,·)) \;\) at \(\; a,b = \) e.g. 1,2