Properties

Label 869.4
Modulus $869$
Conductor $869$
Order $195$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(390))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,40]))
 
pari: [g,chi] = znchar(Mod(4,869))
 

Basic properties

Modulus: \(869\)
Conductor: \(869\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(195\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 869.bc

\(\chi_{869}(4,\cdot)\) \(\chi_{869}(5,\cdot)\) \(\chi_{869}(9,\cdot)\) \(\chi_{869}(16,\cdot)\) \(\chi_{869}(20,\cdot)\) \(\chi_{869}(25,\cdot)\) \(\chi_{869}(26,\cdot)\) \(\chi_{869}(31,\cdot)\) \(\chi_{869}(36,\cdot)\) \(\chi_{869}(42,\cdot)\) \(\chi_{869}(49,\cdot)\) \(\chi_{869}(81,\cdot)\) \(\chi_{869}(92,\cdot)\) \(\chi_{869}(104,\cdot)\) \(\chi_{869}(115,\cdot)\) \(\chi_{869}(119,\cdot)\) \(\chi_{869}(124,\cdot)\) \(\chi_{869}(130,\cdot)\) \(\chi_{869}(152,\cdot)\) \(\chi_{869}(163,\cdot)\) \(\chi_{869}(169,\cdot)\) \(\chi_{869}(174,\cdot)\) \(\chi_{869}(190,\cdot)\) \(\chi_{869}(202,\cdot)\) \(\chi_{869}(203,\cdot)\) \(\chi_{869}(207,\cdot)\) \(\chi_{869}(234,\cdot)\) \(\chi_{869}(246,\cdot)\) \(\chi_{869}(256,\cdot)\) \(\chi_{869}(257,\cdot)\) ...

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: $\Q(\zeta_{195})$
Fixed field: Number field defined by a degree 195 polynomial (not computed)

Values on generators

\((475,793)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{4}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 869 }(4, a) \) \(1\)\(1\)\(e\left(\frac{119}{195}\right)\)\(e\left(\frac{137}{195}\right)\)\(e\left(\frac{43}{195}\right)\)\(e\left(\frac{31}{195}\right)\)\(e\left(\frac{61}{195}\right)\)\(e\left(\frac{163}{195}\right)\)\(e\left(\frac{54}{65}\right)\)\(e\left(\frac{79}{195}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{12}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 869 }(4,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 869 }(4,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 869 }(4,·),\chi_{ 869 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 869 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2