Properties

Label 810.43
Modulus $810$
Conductor $405$
Order $108$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,81]))
 
pari: [g,chi] = znchar(Mod(43,810))
 

Basic properties

Modulus: \(810\)
Conductor: \(405\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{405}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 810.x

\(\chi_{810}(7,\cdot)\) \(\chi_{810}(13,\cdot)\) \(\chi_{810}(43,\cdot)\) \(\chi_{810}(67,\cdot)\) \(\chi_{810}(97,\cdot)\) \(\chi_{810}(103,\cdot)\) \(\chi_{810}(133,\cdot)\) \(\chi_{810}(157,\cdot)\) \(\chi_{810}(187,\cdot)\) \(\chi_{810}(193,\cdot)\) \(\chi_{810}(223,\cdot)\) \(\chi_{810}(247,\cdot)\) \(\chi_{810}(277,\cdot)\) \(\chi_{810}(283,\cdot)\) \(\chi_{810}(313,\cdot)\) \(\chi_{810}(337,\cdot)\) \(\chi_{810}(367,\cdot)\) \(\chi_{810}(373,\cdot)\) \(\chi_{810}(403,\cdot)\) \(\chi_{810}(427,\cdot)\) \(\chi_{810}(457,\cdot)\) \(\chi_{810}(463,\cdot)\) \(\chi_{810}(493,\cdot)\) \(\chi_{810}(517,\cdot)\) \(\chi_{810}(547,\cdot)\) \(\chi_{810}(553,\cdot)\) \(\chi_{810}(583,\cdot)\) \(\chi_{810}(607,\cdot)\) \(\chi_{810}(637,\cdot)\) \(\chi_{810}(643,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((731,487)\) → \((e\left(\frac{11}{27}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 810 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{29}{108}\right)\)\(e\left(\frac{8}{27}\right)\)\(e\left(\frac{55}{108}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{79}{108}\right)\)\(e\left(\frac{31}{54}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{16}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 810 }(43,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 810 }(43,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 810 }(43,·),\chi_{ 810 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 810 }(43,·)) \;\) at \(\; a,b = \) e.g. 1,2