Properties

Label 111.76
Modulus $111$
Conductor $37$
Order $36$
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(111, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(76,111))
 

Basic properties

Modulus: \(111\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{37}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 111.r

\(\chi_{111}(13,\cdot)\) \(\chi_{111}(19,\cdot)\) \(\chi_{111}(22,\cdot)\) \(\chi_{111}(52,\cdot)\) \(\chi_{111}(55,\cdot)\) \(\chi_{111}(61,\cdot)\) \(\chi_{111}(76,\cdot)\) \(\chi_{111}(79,\cdot)\) \(\chi_{111}(91,\cdot)\) \(\chi_{111}(94,\cdot)\) \(\chi_{111}(106,\cdot)\) \(\chi_{111}(109,\cdot)\)

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: \(\Q(\zeta_{37})\)

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 111 }(76,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 111 }(76,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 111 }(76,·),\chi_{ 111 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 111 }(76,·)) \;\) at \(\; a,b = \) e.g. 1,2