Properties

Label 861.17
Modulus $861$
Conductor $861$
Order $120$
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(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([60,20,99]))
 
pari: [g,chi] = znchar(Mod(17,861))
 

Basic properties

Modulus: \(861\)
Conductor: \(861\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
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.cj

\(\chi_{861}(17,\cdot)\) \(\chi_{861}(26,\cdot)\) \(\chi_{861}(47,\cdot)\) \(\chi_{861}(89,\cdot)\) \(\chi_{861}(101,\cdot)\) \(\chi_{861}(110,\cdot)\) \(\chi_{861}(152,\cdot)\) \(\chi_{861}(194,\cdot)\) \(\chi_{861}(227,\cdot)\) \(\chi_{861}(257,\cdot)\) \(\chi_{861}(299,\cdot)\) \(\chi_{861}(311,\cdot)\) \(\chi_{861}(341,\cdot)\) \(\chi_{861}(362,\cdot)\) \(\chi_{861}(395,\cdot)\) \(\chi_{861}(404,\cdot)\) \(\chi_{861}(416,\cdot)\) \(\chi_{861}(425,\cdot)\) \(\chi_{861}(458,\cdot)\) \(\chi_{861}(479,\cdot)\) \(\chi_{861}(509,\cdot)\) \(\chi_{861}(521,\cdot)\) \(\chi_{861}(563,\cdot)\) \(\chi_{861}(593,\cdot)\) \(\chi_{861}(626,\cdot)\) \(\chi_{861}(668,\cdot)\) \(\chi_{861}(710,\cdot)\) \(\chi_{861}(719,\cdot)\) \(\chi_{861}(731,\cdot)\) \(\chi_{861}(773,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((575,493,211)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{33}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 861 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{77}{120}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{107}{120}\right)\)\(e\left(\frac{31}{120}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 861 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 861 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 861 }(17,·),\chi_{ 861 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 861 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2