Properties

Label 1859.100
Modulus $1859$
Conductor $169$
Order $39$
Real no
Primitive no
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(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10]))
 
pari: [g,chi] = znchar(Mod(100,1859))
 

Basic properties

Modulus: \(1859\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(100,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1859.ba

\(\chi_{1859}(100,\cdot)\) \(\chi_{1859}(133,\cdot)\) \(\chi_{1859}(243,\cdot)\) \(\chi_{1859}(276,\cdot)\) \(\chi_{1859}(386,\cdot)\) \(\chi_{1859}(419,\cdot)\) \(\chi_{1859}(562,\cdot)\) \(\chi_{1859}(672,\cdot)\) \(\chi_{1859}(705,\cdot)\) \(\chi_{1859}(815,\cdot)\) \(\chi_{1859}(848,\cdot)\) \(\chi_{1859}(958,\cdot)\) \(\chi_{1859}(1101,\cdot)\) \(\chi_{1859}(1134,\cdot)\) \(\chi_{1859}(1244,\cdot)\) \(\chi_{1859}(1277,\cdot)\) \(\chi_{1859}(1387,\cdot)\) \(\chi_{1859}(1420,\cdot)\) \(\chi_{1859}(1530,\cdot)\) \(\chi_{1859}(1563,\cdot)\) \(\chi_{1859}(1673,\cdot)\) \(\chi_{1859}(1706,\cdot)\) \(\chi_{1859}(1816,\cdot)\) \(\chi_{1859}(1849,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: 39.39.27027636582498189040621249864144468324898507852136260989871841246090732111847218889.1

Values on generators

\((508,1354)\) → \((1,e\left(\frac{5}{39}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1859 }(100, a) \) \(1\)\(1\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{2}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1859 }(100,a) \;\) at \(\;a = \) e.g. 2