Properties

Label 861.23
Modulus $861$
Conductor $861$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,10,27]))
 
pari: [g,chi] = znchar(Mod(23,861))
 

Basic properties

Modulus: \(861\)
Conductor: \(861\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 861.bu

\(\chi_{861}(23,\cdot)\) \(\chi_{861}(86,\cdot)\) \(\chi_{861}(107,\cdot)\) \(\chi_{861}(359,\cdot)\) \(\chi_{861}(515,\cdot)\) \(\chi_{861}(578,\cdot)\) \(\chi_{861}(599,\cdot)\) \(\chi_{861}(851,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((575,493,211)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 861 }(23, a) \) \(-1\)\(1\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{23}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 861 }(23,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 861 }(23,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 861 }(23,·),\chi_{ 861 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 861 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2