Properties

Label 8470.1193
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([495,110,624]))
 
pari: [g,chi] = znchar(Mod(1193,8470))
 

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 \(\chi_{4235}(1193,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.do

\(\chi_{8470}(47,\cdot)\) \(\chi_{8470}(103,\cdot)\) \(\chi_{8470}(157,\cdot)\) \(\chi_{8470}(213,\cdot)\) \(\chi_{8470}(257,\cdot)\) \(\chi_{8470}(313,\cdot)\) \(\chi_{8470}(367,\cdot)\) \(\chi_{8470}(383,\cdot)\) \(\chi_{8470}(423,\cdot)\) \(\chi_{8470}(467,\cdot)\) \(\chi_{8470}(537,\cdot)\) \(\chi_{8470}(577,\cdot)\) \(\chi_{8470}(647,\cdot)\) \(\chi_{8470}(663,\cdot)\) \(\chi_{8470}(773,\cdot)\) \(\chi_{8470}(817,\cdot)\) \(\chi_{8470}(873,\cdot)\) \(\chi_{8470}(927,\cdot)\) \(\chi_{8470}(983,\cdot)\) \(\chi_{8470}(1027,\cdot)\) \(\chi_{8470}(1083,\cdot)\) \(\chi_{8470}(1137,\cdot)\) \(\chi_{8470}(1153,\cdot)\) \(\chi_{8470}(1193,\cdot)\) \(\chi_{8470}(1263,\cdot)\) \(\chi_{8470}(1307,\cdot)\) \(\chi_{8470}(1347,\cdot)\) \(\chi_{8470}(1417,\cdot)\) \(\chi_{8470}(1433,\cdot)\) \(\chi_{8470}(1543,\cdot)\) ...

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

\((6777,6051,7141)\) → \((-i,e\left(\frac{1}{6}\right),e\left(\frac{52}{55}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{53}{220}\right)\)\(e\left(\frac{161}{660}\right)\)\(e\left(\frac{133}{165}\right)\)\(e\left(\frac{101}{132}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{63}{110}\right)\)\(e\left(\frac{157}{330}\right)\)\(e\left(\frac{523}{660}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(1193,a) \;\) at \(\;a = \) e.g. 2