Properties

Label 1617.53
Modulus $1617$
Conductor $1617$
Order $210$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,50,126]))
 
pari: [g,chi] = znchar(Mod(53,1617))
 

Basic properties

Modulus: \(1617\)
Conductor: \(1617\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1617.cl

\(\chi_{1617}(53,\cdot)\) \(\chi_{1617}(86,\cdot)\) \(\chi_{1617}(137,\cdot)\) \(\chi_{1617}(158,\cdot)\) \(\chi_{1617}(170,\cdot)\) \(\chi_{1617}(179,\cdot)\) \(\chi_{1617}(191,\cdot)\) \(\chi_{1617}(212,\cdot)\) \(\chi_{1617}(284,\cdot)\) \(\chi_{1617}(317,\cdot)\) \(\chi_{1617}(368,\cdot)\) \(\chi_{1617}(389,\cdot)\) \(\chi_{1617}(401,\cdot)\) \(\chi_{1617}(443,\cdot)\) \(\chi_{1617}(515,\cdot)\) \(\chi_{1617}(548,\cdot)\) \(\chi_{1617}(599,\cdot)\) \(\chi_{1617}(620,\cdot)\) \(\chi_{1617}(632,\cdot)\) \(\chi_{1617}(641,\cdot)\) \(\chi_{1617}(653,\cdot)\) \(\chi_{1617}(674,\cdot)\) \(\chi_{1617}(746,\cdot)\) \(\chi_{1617}(779,\cdot)\) \(\chi_{1617}(830,\cdot)\) \(\chi_{1617}(872,\cdot)\) \(\chi_{1617}(884,\cdot)\) \(\chi_{1617}(905,\cdot)\) \(\chi_{1617}(977,\cdot)\) \(\chi_{1617}(1061,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((1079,199,442)\) → \((-1,e\left(\frac{5}{21}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 1617 }(53, a) \) \(-1\)\(1\)\(e\left(\frac{61}{210}\right)\)\(e\left(\frac{61}{105}\right)\)\(e\left(\frac{169}{210}\right)\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{17}{105}\right)\)\(e\left(\frac{179}{210}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{27}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1617 }(53,a) \;\) at \(\;a = \) e.g. 2