Properties

Label 810.731
Modulus $810$
Conductor $81$
Order $54$
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(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,0]))
 
pari: [g,chi] = znchar(Mod(731,810))
 

Basic properties

Modulus: \(810\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{81}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 810.u

\(\chi_{810}(11,\cdot)\) \(\chi_{810}(41,\cdot)\) \(\chi_{810}(101,\cdot)\) \(\chi_{810}(131,\cdot)\) \(\chi_{810}(191,\cdot)\) \(\chi_{810}(221,\cdot)\) \(\chi_{810}(281,\cdot)\) \(\chi_{810}(311,\cdot)\) \(\chi_{810}(371,\cdot)\) \(\chi_{810}(401,\cdot)\) \(\chi_{810}(461,\cdot)\) \(\chi_{810}(491,\cdot)\) \(\chi_{810}(551,\cdot)\) \(\chi_{810}(581,\cdot)\) \(\chi_{810}(641,\cdot)\) \(\chi_{810}(671,\cdot)\) \(\chi_{810}(731,\cdot)\) \(\chi_{810}(761,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((731,487)\) → \((e\left(\frac{1}{54}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 810 }(731, a) \) \(-1\)\(1\)\(e\left(\frac{8}{27}\right)\)\(e\left(\frac{13}{54}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{11}{54}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{53}{54}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 810 }(731,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 810 }(731,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 810 }(731,·),\chi_{ 810 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 810 }(731,·)) \;\) at \(\; a,b = \) e.g. 1,2