Properties

Label 1859.75
Modulus $1859$
Conductor $1859$
Order $390$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1859, base_ring=CyclotomicField(390))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([234,355]))
 
pari: [g,chi] = znchar(Mod(75,1859))
 

Basic properties

Modulus: \(1859\)
Conductor: \(1859\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(390\)
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.bs

\(\chi_{1859}(4,\cdot)\) \(\chi_{1859}(36,\cdot)\) \(\chi_{1859}(49,\cdot)\) \(\chi_{1859}(69,\cdot)\) \(\chi_{1859}(75,\cdot)\) \(\chi_{1859}(82,\cdot)\) \(\chi_{1859}(108,\cdot)\) \(\chi_{1859}(114,\cdot)\) \(\chi_{1859}(179,\cdot)\) \(\chi_{1859}(212,\cdot)\) \(\chi_{1859}(218,\cdot)\) \(\chi_{1859}(225,\cdot)\) \(\chi_{1859}(251,\cdot)\) \(\chi_{1859}(257,\cdot)\) \(\chi_{1859}(290,\cdot)\) \(\chi_{1859}(322,\cdot)\) \(\chi_{1859}(335,\cdot)\) \(\chi_{1859}(355,\cdot)\) \(\chi_{1859}(368,\cdot)\) \(\chi_{1859}(394,\cdot)\) \(\chi_{1859}(400,\cdot)\) \(\chi_{1859}(433,\cdot)\) \(\chi_{1859}(465,\cdot)\) \(\chi_{1859}(478,\cdot)\) \(\chi_{1859}(498,\cdot)\) \(\chi_{1859}(504,\cdot)\) \(\chi_{1859}(511,\cdot)\) \(\chi_{1859}(537,\cdot)\) \(\chi_{1859}(543,\cdot)\) \(\chi_{1859}(576,\cdot)\) ...

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: $\Q(\zeta_{195})$
Fixed field: Number field defined by a degree 390 polynomial (not computed)

Values on generators

\((508,1354)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{71}{78}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1859 }(75, a) \) \(1\)\(1\)\(e\left(\frac{199}{390}\right)\)\(e\left(\frac{131}{195}\right)\)\(e\left(\frac{4}{195}\right)\)\(e\left(\frac{77}{130}\right)\)\(e\left(\frac{71}{390}\right)\)\(e\left(\frac{233}{390}\right)\)\(e\left(\frac{69}{130}\right)\)\(e\left(\frac{67}{195}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{9}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1859 }(75,a) \;\) at \(\;a = \) e.g. 2