Properties

Label 8470.1783
Modulus $8470$
Conductor $4235$
Order $132$
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(132))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([99,110,72]))
 
pari: [g,chi] = znchar(Mod(1783,8470))
 

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 \(\chi_{4235}(1783,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.db

\(\chi_{8470}(353,\cdot)\) \(\chi_{8470}(397,\cdot)\) \(\chi_{8470}(507,\cdot)\) \(\chi_{8470}(1013,\cdot)\) \(\chi_{8470}(1123,\cdot)\) \(\chi_{8470}(1167,\cdot)\) \(\chi_{8470}(1277,\cdot)\) \(\chi_{8470}(1783,\cdot)\) \(\chi_{8470}(1893,\cdot)\) \(\chi_{8470}(2047,\cdot)\) \(\chi_{8470}(2553,\cdot)\) \(\chi_{8470}(2707,\cdot)\) \(\chi_{8470}(2817,\cdot)\) \(\chi_{8470}(3323,\cdot)\) \(\chi_{8470}(3433,\cdot)\) \(\chi_{8470}(3477,\cdot)\) \(\chi_{8470}(3587,\cdot)\) \(\chi_{8470}(4093,\cdot)\) \(\chi_{8470}(4203,\cdot)\) \(\chi_{8470}(4247,\cdot)\) \(\chi_{8470}(4863,\cdot)\) \(\chi_{8470}(4973,\cdot)\) \(\chi_{8470}(5017,\cdot)\) \(\chi_{8470}(5127,\cdot)\) \(\chi_{8470}(5633,\cdot)\) \(\chi_{8470}(5743,\cdot)\) \(\chi_{8470}(5787,\cdot)\) \(\chi_{8470}(5897,\cdot)\) \(\chi_{8470}(6403,\cdot)\) \(\chi_{8470}(6513,\cdot)\) ...

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

Values on generators

\((6777,6051,7141)\) → \((-i,e\left(\frac{5}{6}\right),e\left(\frac{6}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{41}{132}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{13}{132}\right)\)\(i\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{43}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(1783,a) \;\) at \(\;a = \) e.g. 2