Properties

Label 8470.8249
Modulus $8470$
Conductor $4235$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,11,57]))
 
pari: [g,chi] = znchar(Mod(8249,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(4014,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cm

\(\chi_{8470}(439,\cdot)\) \(\chi_{8470}(549,\cdot)\) \(\chi_{8470}(1319,\cdot)\) \(\chi_{8470}(1979,\cdot)\) \(\chi_{8470}(2089,\cdot)\) \(\chi_{8470}(2749,\cdot)\) \(\chi_{8470}(2859,\cdot)\) \(\chi_{8470}(3519,\cdot)\) \(\chi_{8470}(4289,\cdot)\) \(\chi_{8470}(4399,\cdot)\) \(\chi_{8470}(5059,\cdot)\) \(\chi_{8470}(5169,\cdot)\) \(\chi_{8470}(5829,\cdot)\) \(\chi_{8470}(5939,\cdot)\) \(\chi_{8470}(6599,\cdot)\) \(\chi_{8470}(6709,\cdot)\) \(\chi_{8470}(7369,\cdot)\) \(\chi_{8470}(7479,\cdot)\) \(\chi_{8470}(8139,\cdot)\) \(\chi_{8470}(8249,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((6777,6051,7141)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(8249, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{19}{66}\right)\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{7}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(8249,a) \;\) at \(\;a = \) e.g. 2