Properties

Label 1870.41
Modulus $1870$
Conductor $187$
Order $80$
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(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,24,55]))
 
pari: [g,chi] = znchar(Mod(41,1870))
 

Basic properties

Modulus: \(1870\)
Conductor: \(187\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{187}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1870.co

\(\chi_{1870}(41,\cdot)\) \(\chi_{1870}(61,\cdot)\) \(\chi_{1870}(211,\cdot)\) \(\chi_{1870}(261,\cdot)\) \(\chi_{1870}(371,\cdot)\) \(\chi_{1870}(381,\cdot)\) \(\chi_{1870}(431,\cdot)\) \(\chi_{1870}(481,\cdot)\) \(\chi_{1870}(541,\cdot)\) \(\chi_{1870}(601,\cdot)\) \(\chi_{1870}(651,\cdot)\) \(\chi_{1870}(711,\cdot)\) \(\chi_{1870}(721,\cdot)\) \(\chi_{1870}(811,\cdot)\) \(\chi_{1870}(821,\cdot)\) \(\chi_{1870}(921,\cdot)\) \(\chi_{1870}(941,\cdot)\) \(\chi_{1870}(981,\cdot)\) \(\chi_{1870}(1031,\cdot)\) \(\chi_{1870}(1051,\cdot)\) \(\chi_{1870}(1091,\cdot)\) \(\chi_{1870}(1151,\cdot)\) \(\chi_{1870}(1161,\cdot)\) \(\chi_{1870}(1201,\cdot)\) \(\chi_{1870}(1251,\cdot)\) \(\chi_{1870}(1261,\cdot)\) \(\chi_{1870}(1371,\cdot)\) \(\chi_{1870}(1421,\cdot)\) \(\chi_{1870}(1491,\cdot)\) \(\chi_{1870}(1591,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((1497,1531,1431)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{11}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 1870 }(41, a) \) \(1\)\(1\)\(e\left(\frac{7}{80}\right)\)\(e\left(\frac{53}{80}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{21}{40}\right)\)\(-i\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{21}{80}\right)\)\(e\left(\frac{3}{80}\right)\)\(e\left(\frac{79}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1870 }(41,a) \;\) at \(\;a = \) e.g. 2