Properties

Label 1870.81
Modulus $1870$
Conductor $187$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1870, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,4,5]))
 
pari: [g,chi] = znchar(Mod(81,1870))
 

Basic properties

Modulus: \(1870\)
Conductor: \(187\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{187}(81,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1870.bv

\(\chi_{1870}(81,\cdot)\) \(\chi_{1870}(191,\cdot)\) \(\chi_{1870}(251,\cdot)\) \(\chi_{1870}(361,\cdot)\) \(\chi_{1870}(421,\cdot)\) \(\chi_{1870}(531,\cdot)\) \(\chi_{1870}(1611,\cdot)\) \(\chi_{1870}(1721,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((1497,1531,1431)\) → \((1,e\left(\frac{1}{5}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 1870 }(81, a) \) \(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(1\)\(-i\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1870 }(81,a) \;\) at \(\;a = \) e.g. 2