Properties

Label 927.109
Modulus $927$
Conductor $103$
Order $102$
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(102))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,83]))
 
pari: [g,chi] = znchar(Mod(109,927))
 

Basic properties

Modulus: \(927\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{103}(6,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 927.bk

\(\chi_{927}(109,\cdot)\) \(\chi_{927}(154,\cdot)\) \(\chi_{927}(181,\cdot)\) \(\chi_{927}(190,\cdot)\) \(\chi_{927}(199,\cdot)\) \(\chi_{927}(217,\cdot)\) \(\chi_{927}(226,\cdot)\) \(\chi_{927}(271,\cdot)\) \(\chi_{927}(280,\cdot)\) \(\chi_{927}(307,\cdot)\) \(\chi_{927}(352,\cdot)\) \(\chi_{927}(379,\cdot)\) \(\chi_{927}(397,\cdot)\) \(\chi_{927}(424,\cdot)\) \(\chi_{927}(433,\cdot)\) \(\chi_{927}(460,\cdot)\) \(\chi_{927}(487,\cdot)\) \(\chi_{927}(496,\cdot)\) \(\chi_{927}(550,\cdot)\) \(\chi_{927}(559,\cdot)\) \(\chi_{927}(568,\cdot)\) \(\chi_{927}(577,\cdot)\) \(\chi_{927}(586,\cdot)\) \(\chi_{927}(658,\cdot)\) \(\chi_{927}(685,\cdot)\) \(\chi_{927}(703,\cdot)\) \(\chi_{927}(766,\cdot)\) \(\chi_{927}(775,\cdot)\) \(\chi_{927}(820,\cdot)\) \(\chi_{927}(829,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((722,829)\) → \((1,e\left(\frac{83}{102}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{41}{51}\right)\)\(e\left(\frac{31}{51}\right)\)\(e\left(\frac{83}{102}\right)\)\(e\left(\frac{13}{51}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{65}{102}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{11}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 927 }(109,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 927 }(109,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 927 }(109,·),\chi_{ 927 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 927 }(109,·)) \;\) at \(\; a,b = \) e.g. 1,2