Properties

Label 8470.53
Modulus $8470$
Conductor $4235$
Order $660$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, base_ring=CyclotomicField(660))
 
M = H._module
 
chi = DirichletCharacter(H, M([495,440,636]))
 
pari: [g,chi] = znchar(Mod(53,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(660\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.dq

\(\chi_{8470}(37,\cdot)\) \(\chi_{8470}(53,\cdot)\) \(\chi_{8470}(93,\cdot)\) \(\chi_{8470}(137,\cdot)\) \(\chi_{8470}(163,\cdot)\) \(\chi_{8470}(207,\cdot)\) \(\chi_{8470}(247,\cdot)\) \(\chi_{8470}(317,\cdot)\) \(\chi_{8470}(333,\cdot)\) \(\chi_{8470}(443,\cdot)\) \(\chi_{8470}(543,\cdot)\) \(\chi_{8470}(597,\cdot)\) \(\chi_{8470}(653,\cdot)\) \(\chi_{8470}(697,\cdot)\) \(\chi_{8470}(823,\cdot)\) \(\chi_{8470}(863,\cdot)\) \(\chi_{8470}(907,\cdot)\) \(\chi_{8470}(933,\cdot)\) \(\chi_{8470}(1017,\cdot)\) \(\chi_{8470}(1087,\cdot)\) \(\chi_{8470}(1103,\cdot)\) \(\chi_{8470}(1257,\cdot)\) \(\chi_{8470}(1313,\cdot)\) \(\chi_{8470}(1367,\cdot)\) \(\chi_{8470}(1423,\cdot)\) \(\chi_{8470}(1467,\cdot)\) \(\chi_{8470}(1523,\cdot)\)

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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

Values on generators

\((6777,6051,7141)\) → \((-i,e\left(\frac{2}{3}\right),e\left(\frac{53}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(53, a) \) \(-1\)\(1\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{127}{220}\right)\)\(e\left(\frac{419}{660}\right)\)\(e\left(\frac{269}{330}\right)\)\(e\left(\frac{5}{132}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{97}{110}\right)\)\(e\left(\frac{89}{165}\right)\)\(e\left(\frac{367}{660}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(53,a) \;\) at \(\;a = \) e.g. 2