Properties

Label 320.37
Modulus $320$
Conductor $320$
Order $16$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

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([0,9,4]))
 
pari: [g,chi] = znchar(Mod(37,320))
 

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.bi

\(\chi_{320}(13,\cdot)\) \(\chi_{320}(37,\cdot)\) \(\chi_{320}(93,\cdot)\) \(\chi_{320}(117,\cdot)\) \(\chi_{320}(173,\cdot)\) \(\chi_{320}(197,\cdot)\) \(\chi_{320}(253,\cdot)\) \(\chi_{320}(277,\cdot)\)

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: 16.0.147573952589676412928000000000000.2

Values on generators

\((191,261,257)\) → \((1,e\left(\frac{9}{16}\right),i)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 320 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(1\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 320 }(37,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 320 }(37,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 320 }(37,·),\chi_{ 320 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 320 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2