Properties

Label 1859.864
Modulus $1859$
Conductor $143$
Order $60$
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(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([54,25]))
 
pari: [g,chi] = znchar(Mod(864,1859))
 

Basic properties

Modulus: \(1859\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{143}(6,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1859.bd

\(\chi_{1859}(19,\cdot)\) \(\chi_{1859}(150,\cdot)\) \(\chi_{1859}(249,\cdot)\) \(\chi_{1859}(596,\cdot)\) \(\chi_{1859}(657,\cdot)\) \(\chi_{1859}(695,\cdot)\) \(\chi_{1859}(756,\cdot)\) \(\chi_{1859}(765,\cdot)\) \(\chi_{1859}(864,\cdot)\) \(\chi_{1859}(1272,\cdot)\) \(\chi_{1859}(1333,\cdot)\) \(\chi_{1859}(1371,\cdot)\) \(\chi_{1859}(1432,\cdot)\) \(\chi_{1859}(1502,\cdot)\) \(\chi_{1859}(1601,\cdot)\) \(\chi_{1859}(1779,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((508,1354)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{5}{12}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1859 }(864, a) \) \(1\)\(1\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1859 }(864,a) \;\) at \(\;a = \) e.g. 2