Properties

Label 1859.1037
Modulus $1859$
Conductor $143$
Order $30$
Real no
Primitive no
Minimal no
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(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([24,25]))
 
pari: [g,chi] = znchar(Mod(1037,1859))
 

Basic properties

Modulus: \(1859\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{143}(36,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1859.y

\(\chi_{1859}(147,\cdot)\) \(\chi_{1859}(192,\cdot)\) \(\chi_{1859}(361,\cdot)\) \(\chi_{1859}(654,\cdot)\) \(\chi_{1859}(823,\cdot)\) \(\chi_{1859}(1037,\cdot)\) \(\chi_{1859}(1499,\cdot)\) \(\chi_{1859}(1544,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.69503752297329754905479727341904896738456941915804813.1

Values on generators

\((508,1354)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{5}{6}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1859 }(1037, a) \) \(1\)\(1\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1859 }(1037,a) \;\) at \(\;a = \) e.g. 2