Properties

Label 8470.373
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([99,44,90]))
 
pari: [g,chi] = znchar(Mod(373,8470))
 

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 \(\chi_{4235}(373,\cdot)\)
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()
 
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

\((6777,6051,7141)\) → \((-i,e\left(\frac{1}{3}\right),e\left(\frac{15}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{65}{132}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{85}{132}\right)\)\(-i\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{7}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(373,a) \;\) at \(\;a = \) e.g. 2