Properties

Label 1859.12
Modulus $1859$
Conductor $169$
Order $26$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1859, base_ring=CyclotomicField(26))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,21]))
 
pari: [g,chi] = znchar(Mod(12,1859))
 

Basic properties

Modulus: \(1859\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(12,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1859.w

\(\chi_{1859}(12,\cdot)\) \(\chi_{1859}(155,\cdot)\) \(\chi_{1859}(298,\cdot)\) \(\chi_{1859}(441,\cdot)\) \(\chi_{1859}(584,\cdot)\) \(\chi_{1859}(727,\cdot)\) \(\chi_{1859}(870,\cdot)\) \(\chi_{1859}(1156,\cdot)\) \(\chi_{1859}(1299,\cdot)\) \(\chi_{1859}(1442,\cdot)\) \(\chi_{1859}(1585,\cdot)\) \(\chi_{1859}(1728,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: 26.26.3830224792147131369362629348887201408953937846517364173.1

Values on generators

\((508,1354)\) → \((1,e\left(\frac{21}{26}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1859 }(12, a) \) \(1\)\(1\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{10}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1859 }(12,a) \;\) at \(\;a = \) e.g. 2