Properties

Label 99.68
Modulus $99$
Conductor $99$
Order $30$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(99, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([25,3]))
 
pari: [g,chi] = znchar(Mod(68,99))
 

Basic properties

Modulus: \(99\)
Conductor: \(99\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 99.p

\(\chi_{99}(2,\cdot)\) \(\chi_{99}(29,\cdot)\) \(\chi_{99}(41,\cdot)\) \(\chi_{99}(50,\cdot)\) \(\chi_{99}(68,\cdot)\) \(\chi_{99}(74,\cdot)\) \(\chi_{99}(83,\cdot)\) \(\chi_{99}(95,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: \(\Q(\zeta_{99})^+\)

Values on generators

\((56,46)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{10}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 99 }(68, a) \) \(1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(-1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 99 }(68,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 99 }(68,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 99 }(68,·),\chi_{ 99 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 99 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2