Properties

Label 1859.14
Modulus $1859$
Conductor $1859$
Order $65$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1859, base_ring=CyclotomicField(130))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([104,90]))
 
pari: [g,chi] = znchar(Mod(14,1859))
 

Basic properties

Modulus: \(1859\)
Conductor: \(1859\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(65\)
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 1859.bf

\(\chi_{1859}(14,\cdot)\) \(\chi_{1859}(27,\cdot)\) \(\chi_{1859}(53,\cdot)\) \(\chi_{1859}(92,\cdot)\) \(\chi_{1859}(157,\cdot)\) \(\chi_{1859}(196,\cdot)\) \(\chi_{1859}(235,\cdot)\) \(\chi_{1859}(300,\cdot)\) \(\chi_{1859}(313,\cdot)\) \(\chi_{1859}(378,\cdot)\) \(\chi_{1859}(443,\cdot)\) \(\chi_{1859}(456,\cdot)\) \(\chi_{1859}(482,\cdot)\) \(\chi_{1859}(521,\cdot)\) \(\chi_{1859}(586,\cdot)\) \(\chi_{1859}(599,\cdot)\) \(\chi_{1859}(625,\cdot)\) \(\chi_{1859}(664,\cdot)\) \(\chi_{1859}(729,\cdot)\) \(\chi_{1859}(742,\cdot)\) \(\chi_{1859}(768,\cdot)\) \(\chi_{1859}(807,\cdot)\) \(\chi_{1859}(872,\cdot)\) \(\chi_{1859}(885,\cdot)\) \(\chi_{1859}(911,\cdot)\) \(\chi_{1859}(950,\cdot)\) \(\chi_{1859}(1028,\cdot)\) \(\chi_{1859}(1054,\cdot)\) \(\chi_{1859}(1093,\cdot)\) \(\chi_{1859}(1158,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{65})$
Fixed field: Number field defined by a degree 65 polynomial

Values on generators

\((508,1354)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{9}{13}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1859 }(14, a) \) \(1\)\(1\)\(e\left(\frac{32}{65}\right)\)\(e\left(\frac{16}{65}\right)\)\(e\left(\frac{64}{65}\right)\)\(e\left(\frac{28}{65}\right)\)\(e\left(\frac{48}{65}\right)\)\(e\left(\frac{44}{65}\right)\)\(e\left(\frac{31}{65}\right)\)\(e\left(\frac{32}{65}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{3}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1859 }(14,a) \;\) at \(\;a = \) e.g. 2