Properties

Label 8470.1663
Modulus $8470$
Conductor $4235$
Order $660$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(8470, base_ring=CyclotomicField(660))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([495,440,186]))
 
pari: [g,chi] = znchar(Mod(1663,8470))
 

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 \(\chi_{4235}(1663,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.dp

\(\chi_{8470}(107,\cdot)\) \(\chi_{8470}(123,\cdot)\) \(\chi_{8470}(193,\cdot)\) \(\chi_{8470}(277,\cdot)\) \(\chi_{8470}(303,\cdot)\) \(\chi_{8470}(347,\cdot)\) \(\chi_{8470}(387,\cdot)\) \(\chi_{8470}(513,\cdot)\) \(\chi_{8470}(557,\cdot)\) \(\chi_{8470}(613,\cdot)\) \(\chi_{8470}(667,\cdot)\) \(\chi_{8470}(767,\cdot)\) \(\chi_{8470}(877,\cdot)\) \(\chi_{8470}(893,\cdot)\) \(\chi_{8470}(963,\cdot)\) \(\chi_{8470}(1003,\cdot)\) \(\chi_{8470}(1047,\cdot)\) \(\chi_{8470}(1073,\cdot)\) \(\chi_{8470}(1117,\cdot)\) \(\chi_{8470}(1157,\cdot)\) \(\chi_{8470}(1173,\cdot)\) \(\chi_{8470}(1227,\cdot)\) \(\chi_{8470}(1283,\cdot)\) \(\chi_{8470}(1327,\cdot)\) \(\chi_{8470}(1383,\cdot)\) \(\chi_{8470}(1437,\cdot)\) \(\chi_{8470}(1493,\cdot)\) \(\chi_{8470}(1537,\cdot)\) \(\chi_{8470}(1647,\cdot)\) \(\chi_{8470}(1663,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((6777,6051,7141)\) → \((-i,e\left(\frac{2}{3}\right),e\left(\frac{31}{110}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{157}{220}\right)\)\(e\left(\frac{149}{660}\right)\)\(e\left(\frac{37}{165}\right)\)\(e\left(\frac{41}{132}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{149}{165}\right)\)\(e\left(\frac{607}{660}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(1663,a) \;\) at \(\;a = \) e.g. 2