Properties

Label 2169.17
Modulus $2169$
Conductor $723$
Order $80$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,37]))
 
pari: [g,chi] = znchar(Mod(17,2169))
 

Basic properties

Modulus: \(2169\)
Conductor: \(723\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{723}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2169.df

\(\chi_{2169}(17,\cdot)\) \(\chi_{2169}(26,\cdot)\) \(\chi_{2169}(215,\cdot)\) \(\chi_{2169}(224,\cdot)\) \(\chi_{2169}(269,\cdot)\) \(\chi_{2169}(314,\cdot)\) \(\chi_{2169}(377,\cdot)\) \(\chi_{2169}(449,\cdot)\) \(\chi_{2169}(503,\cdot)\) \(\chi_{2169}(539,\cdot)\) \(\chi_{2169}(575,\cdot)\) \(\chi_{2169}(584,\cdot)\) \(\chi_{2169}(620,\cdot)\) \(\chi_{2169}(638,\cdot)\) \(\chi_{2169}(746,\cdot)\) \(\chi_{2169}(863,\cdot)\) \(\chi_{2169}(1007,\cdot)\) \(\chi_{2169}(1088,\cdot)\) \(\chi_{2169}(1322,\cdot)\) \(\chi_{2169}(1403,\cdot)\) \(\chi_{2169}(1547,\cdot)\) \(\chi_{2169}(1664,\cdot)\) \(\chi_{2169}(1772,\cdot)\) \(\chi_{2169}(1790,\cdot)\) \(\chi_{2169}(1826,\cdot)\) \(\chi_{2169}(1835,\cdot)\) \(\chi_{2169}(1871,\cdot)\) \(\chi_{2169}(1907,\cdot)\) \(\chi_{2169}(1961,\cdot)\) \(\chi_{2169}(2033,\cdot)\) ...

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

\((965,730)\) → \((-1,e\left(\frac{37}{80}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2169 }(17, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{59}{80}\right)\)\(e\left(\frac{67}{80}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2169 }(17,a) \;\) at \(\;a = \) e.g. 2