Properties

Label 1710.1363
Modulus $1710$
Conductor $855$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,27,14]))
 
pari: [g,chi] = znchar(Mod(1363,1710))
 

Basic properties

Modulus: \(1710\)
Conductor: \(855\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{855}(508,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1710.dq

\(\chi_{1710}(193,\cdot)\) \(\chi_{1710}(337,\cdot)\) \(\chi_{1710}(547,\cdot)\) \(\chi_{1710}(553,\cdot)\) \(\chi_{1710}(583,\cdot)\) \(\chi_{1710}(637,\cdot)\) \(\chi_{1710}(877,\cdot)\) \(\chi_{1710}(1237,\cdot)\) \(\chi_{1710}(1267,\cdot)\) \(\chi_{1710}(1363,\cdot)\) \(\chi_{1710}(1573,\cdot)\) \(\chi_{1710}(1663,\cdot)\)

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: 36.36.17849776228866488737715206984999102954438314099226129130939288096733391284942626953125.1

Values on generators

\((191,1027,1351)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{7}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1710 }(1363, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(1\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{4}{9}\right)\)\(-1\)\(i\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{29}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1710 }(1363,a) \;\) at \(\;a = \) e.g. 2