Properties

Label 111.35
Modulus $111$
Conductor $111$
Order $36$
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(111, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18,19]))
 
pari: [g,chi] = znchar(Mod(35,111))
 

Basic properties

Modulus: \(111\)
Conductor: \(111\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 111.q

\(\chi_{111}(2,\cdot)\) \(\chi_{111}(5,\cdot)\) \(\chi_{111}(17,\cdot)\) \(\chi_{111}(20,\cdot)\) \(\chi_{111}(32,\cdot)\) \(\chi_{111}(35,\cdot)\) \(\chi_{111}(50,\cdot)\) \(\chi_{111}(56,\cdot)\) \(\chi_{111}(59,\cdot)\) \(\chi_{111}(89,\cdot)\) \(\chi_{111}(92,\cdot)\) \(\chi_{111}(98,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{111})^+\)

Values on generators

\((38,76)\) → \((-1,e\left(\frac{19}{36}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 111 }(35,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 111 }(35,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 111 }(35,·),\chi_{ 111 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 111 }(35,·)) \;\) at \(\; a,b = \) e.g. 1,2