Properties

Label 847.454
Modulus $847$
Conductor $847$
Order $110$
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(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,108]))
 
pari: [g,chi] = znchar(Mod(454,847))
 

Basic properties

Modulus: \(847\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
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.bb

\(\chi_{847}(20,\cdot)\) \(\chi_{847}(48,\cdot)\) \(\chi_{847}(69,\cdot)\) \(\chi_{847}(97,\cdot)\) \(\chi_{847}(104,\cdot)\) \(\chi_{847}(125,\cdot)\) \(\chi_{847}(146,\cdot)\) \(\chi_{847}(174,\cdot)\) \(\chi_{847}(181,\cdot)\) \(\chi_{847}(223,\cdot)\) \(\chi_{847}(258,\cdot)\) \(\chi_{847}(279,\cdot)\) \(\chi_{847}(300,\cdot)\) \(\chi_{847}(328,\cdot)\) \(\chi_{847}(335,\cdot)\) \(\chi_{847}(356,\cdot)\) \(\chi_{847}(377,\cdot)\) \(\chi_{847}(405,\cdot)\) \(\chi_{847}(412,\cdot)\) \(\chi_{847}(433,\cdot)\) \(\chi_{847}(454,\cdot)\) \(\chi_{847}(482,\cdot)\) \(\chi_{847}(489,\cdot)\) \(\chi_{847}(510,\cdot)\) \(\chi_{847}(531,\cdot)\) \(\chi_{847}(559,\cdot)\) \(\chi_{847}(566,\cdot)\) \(\chi_{847}(587,\cdot)\) \(\chi_{847}(636,\cdot)\) \(\chi_{847}(643,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((122,365)\) → \((-1,e\left(\frac{54}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 847 }(454, a) \) \(-1\)\(1\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{97}{110}\right)\)\(e\left(\frac{52}{55}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{73}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 847 }(454,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 847 }(454,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 847 }(454,·),\chi_{ 847 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 847 }(454,·)) \;\) at \(\; a,b = \) e.g. 1,2