Properties

Label 8712.29
Modulus $8712$
Conductor $8712$
Order $330$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8712, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,165,55,51]))
 
pari: [g,chi] = znchar(Mod(29,8712))
 

Basic properties

Modulus: \(8712\)
Conductor: \(8712\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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 8712.en

\(\chi_{8712}(29,\cdot)\) \(\chi_{8712}(101,\cdot)\) \(\chi_{8712}(149,\cdot)\) \(\chi_{8712}(173,\cdot)\) \(\chi_{8712}(293,\cdot)\) \(\chi_{8712}(365,\cdot)\) \(\chi_{8712}(437,\cdot)\) \(\chi_{8712}(677,\cdot)\) \(\chi_{8712}(821,\cdot)\) \(\chi_{8712}(893,\cdot)\) \(\chi_{8712}(1085,\cdot)\) \(\chi_{8712}(1157,\cdot)\) \(\chi_{8712}(1229,\cdot)\) \(\chi_{8712}(1469,\cdot)\) \(\chi_{8712}(1733,\cdot)\) \(\chi_{8712}(1757,\cdot)\) \(\chi_{8712}(1877,\cdot)\) \(\chi_{8712}(1949,\cdot)\) \(\chi_{8712}(2021,\cdot)\) \(\chi_{8712}(2261,\cdot)\) \(\chi_{8712}(2405,\cdot)\) \(\chi_{8712}(2477,\cdot)\) \(\chi_{8712}(2525,\cdot)\) \(\chi_{8712}(2549,\cdot)\) \(\chi_{8712}(2669,\cdot)\) \(\chi_{8712}(2741,\cdot)\) \(\chi_{8712}(2813,\cdot)\) \(\chi_{8712}(3053,\cdot)\) \(\chi_{8712}(3197,\cdot)\) \(\chi_{8712}(3269,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((6535,4357,1937,5689)\) → \((1,-1,e\left(\frac{1}{6}\right),e\left(\frac{17}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8712 }(29, a) \) \(1\)\(1\)\(e\left(\frac{127}{165}\right)\)\(e\left(\frac{247}{330}\right)\)\(e\left(\frac{73}{165}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{18}{55}\right)\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{89}{165}\right)\)\(e\left(\frac{97}{330}\right)\)\(e\left(\frac{103}{165}\right)\)\(e\left(\frac{57}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8712 }(29,a) \;\) at \(\;a = \) e.g. 2