Properties

Label 8470.417
Modulus $8470$
Conductor $4235$
Order $132$
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(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,88,54]))
 
pari: [g,chi] = znchar(Mod(417,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(417,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.da

\(\chi_{8470}(263,\cdot)\) \(\chi_{8470}(373,\cdot)\) \(\chi_{8470}(417,\cdot)\) \(\chi_{8470}(527,\cdot)\) \(\chi_{8470}(1033,\cdot)\) \(\chi_{8470}(1143,\cdot)\) \(\chi_{8470}(1187,\cdot)\) \(\chi_{8470}(1297,\cdot)\) \(\chi_{8470}(1803,\cdot)\) \(\chi_{8470}(1913,\cdot)\) \(\chi_{8470}(1957,\cdot)\) \(\chi_{8470}(2067,\cdot)\) \(\chi_{8470}(2573,\cdot)\) \(\chi_{8470}(2683,\cdot)\) \(\chi_{8470}(2727,\cdot)\) \(\chi_{8470}(2837,\cdot)\) \(\chi_{8470}(3343,\cdot)\) \(\chi_{8470}(3453,\cdot)\) \(\chi_{8470}(3497,\cdot)\) \(\chi_{8470}(3607,\cdot)\) \(\chi_{8470}(4223,\cdot)\) \(\chi_{8470}(4267,\cdot)\) \(\chi_{8470}(4377,\cdot)\) \(\chi_{8470}(4883,\cdot)\) \(\chi_{8470}(4993,\cdot)\) \(\chi_{8470}(5037,\cdot)\) \(\chi_{8470}(5147,\cdot)\) \(\chi_{8470}(5653,\cdot)\) \(\chi_{8470}(5763,\cdot)\) \(\chi_{8470}(5917,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((6777,6051,7141)\) → \((i,e\left(\frac{2}{3}\right),e\left(\frac{9}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(417, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{127}{132}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{95}{132}\right)\)\(i\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{101}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(417,a) \;\) at \(\;a = \) e.g. 2