Properties

Label 1755.59
Modulus $1755$
Conductor $1755$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1755, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,18,33]))
 
pari: [g,chi] = znchar(Mod(59,1755))
 

Basic properties

Modulus: \(1755\)
Conductor: \(1755\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1755.fa

\(\chi_{1755}(59,\cdot)\) \(\chi_{1755}(119,\cdot)\) \(\chi_{1755}(149,\cdot)\) \(\chi_{1755}(479,\cdot)\) \(\chi_{1755}(644,\cdot)\) \(\chi_{1755}(704,\cdot)\) \(\chi_{1755}(734,\cdot)\) \(\chi_{1755}(1064,\cdot)\) \(\chi_{1755}(1229,\cdot)\) \(\chi_{1755}(1289,\cdot)\) \(\chi_{1755}(1319,\cdot)\) \(\chi_{1755}(1649,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((326,352,1081)\) → \((e\left(\frac{5}{18}\right),-1,e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 1755 }(59, a) \) \(1\)\(1\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(-1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{13}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1755 }(59,a) \;\) at \(\;a = \) e.g. 2