Properties

Label 847.38
Modulus $847$
Conductor $847$
Order $330$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,252]))
 
pari: [g,chi] = znchar(Mod(38,847))
 

Basic properties

Modulus: \(847\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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 847.bd

\(\chi_{847}(5,\cdot)\) \(\chi_{847}(26,\cdot)\) \(\chi_{847}(31,\cdot)\) \(\chi_{847}(38,\cdot)\) \(\chi_{847}(47,\cdot)\) \(\chi_{847}(59,\cdot)\) \(\chi_{847}(75,\cdot)\) \(\chi_{847}(80,\cdot)\) \(\chi_{847}(82,\cdot)\) \(\chi_{847}(103,\cdot)\) \(\chi_{847}(108,\cdot)\) \(\chi_{847}(115,\cdot)\) \(\chi_{847}(136,\cdot)\) \(\chi_{847}(152,\cdot)\) \(\chi_{847}(157,\cdot)\) \(\chi_{847}(159,\cdot)\) \(\chi_{847}(180,\cdot)\) \(\chi_{847}(185,\cdot)\) \(\chi_{847}(192,\cdot)\) \(\chi_{847}(201,\cdot)\) \(\chi_{847}(213,\cdot)\) \(\chi_{847}(229,\cdot)\) \(\chi_{847}(234,\cdot)\) \(\chi_{847}(236,\cdot)\) \(\chi_{847}(257,\cdot)\) \(\chi_{847}(262,\cdot)\) \(\chi_{847}(278,\cdot)\) \(\chi_{847}(290,\cdot)\) \(\chi_{847}(306,\cdot)\) \(\chi_{847}(311,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((122,365)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{42}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 847 }(38, a) \) \(-1\)\(1\)\(e\left(\frac{16}{165}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{32}{165}\right)\)\(e\left(\frac{113}{330}\right)\)\(e\left(\frac{51}{110}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{69}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 847 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 847 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 847 }(38,·),\chi_{ 847 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 847 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2