Properties

Label 917415.777926
Modulus $917415$
Conductor $171$
Order $18$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(917415, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([3,0,8,0,0]))
 
pari: [g,chi] = znchar(Mod(777926,917415))
 

Basic properties

Modulus: \(917415\)
Conductor: \(171\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{171}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 917415.dld

\(\chi_{917415}(69746,\cdot)\) \(\chi_{917415}(295076,\cdot)\) \(\chi_{917415}(584786,\cdot)\) \(\chi_{917415}(745736,\cdot)\) \(\chi_{917415}(777926,\cdot)\) \(\chi_{917415}(842306,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((101936,366967,772561,221446,371926)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{4}{9}\right),1,1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(22\)
\( \chi_{ 917415 }(777926, a) \) \(-1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(-1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 917415 }(777926,a) \;\) at \(\;a = \) e.g. 2