Properties

Label 927.73
Modulus $927$
Conductor $103$
Order $34$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(927, base_ring=CyclotomicField(34))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11]))
 
pari: [g,chi] = znchar(Mod(73,927))
 

Basic properties

Modulus: \(927\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{103}(73,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 927.v

\(\chi_{927}(10,\cdot)\) \(\chi_{927}(37,\cdot)\) \(\chi_{927}(73,\cdot)\) \(\chi_{927}(127,\cdot)\) \(\chi_{927}(145,\cdot)\) \(\chi_{927}(172,\cdot)\) \(\chi_{927}(415,\cdot)\) \(\chi_{927}(451,\cdot)\) \(\chi_{927}(595,\cdot)\) \(\chi_{927}(604,\cdot)\) \(\chi_{927}(640,\cdot)\) \(\chi_{927}(649,\cdot)\) \(\chi_{927}(712,\cdot)\) \(\chi_{927}(748,\cdot)\) \(\chi_{927}(811,\cdot)\) \(\chi_{927}(919,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: 34.0.2652335238355663972863781109929452800183143879582476922663961790823.1

Values on generators

\((722,829)\) → \((1,e\left(\frac{11}{34}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{19}{34}\right)\)\(e\left(\frac{25}{34}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{16}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 927 }(73,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 927 }(73,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 927 }(73,·),\chi_{ 927 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 927 }(73,·)) \;\) at \(\; a,b = \) e.g. 1,2