Properties

Label 8470.191
Modulus $8470$
Conductor $847$
Order $165$
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(330))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,110,246]))
 
pari: [g,chi] = znchar(Mod(191,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(165\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(191,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.dc

\(\chi_{8470}(191,\cdot)\) \(\chi_{8470}(291,\cdot)\) \(\chi_{8470}(361,\cdot)\) \(\chi_{8470}(401,\cdot)\) \(\chi_{8470}(471,\cdot)\) \(\chi_{8470}(641,\cdot)\) \(\chi_{8470}(751,\cdot)\) \(\chi_{8470}(851,\cdot)\) \(\chi_{8470}(961,\cdot)\) \(\chi_{8470}(1061,\cdot)\) \(\chi_{8470}(1131,\cdot)\) \(\chi_{8470}(1171,\cdot)\) \(\chi_{8470}(1241,\cdot)\) \(\chi_{8470}(1411,\cdot)\) \(\chi_{8470}(1521,\cdot)\) \(\chi_{8470}(1621,\cdot)\) \(\chi_{8470}(1731,\cdot)\) \(\chi_{8470}(1831,\cdot)\) \(\chi_{8470}(1901,\cdot)\) \(\chi_{8470}(1941,\cdot)\) \(\chi_{8470}(2011,\cdot)\) \(\chi_{8470}(2291,\cdot)\) \(\chi_{8470}(2391,\cdot)\) \(\chi_{8470}(2601,\cdot)\) \(\chi_{8470}(2711,\cdot)\) \(\chi_{8470}(2781,\cdot)\) \(\chi_{8470}(2951,\cdot)\) \(\chi_{8470}(3061,\cdot)\) \(\chi_{8470}(3161,\cdot)\) \(\chi_{8470}(3271,\cdot)\) ...

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

Values on generators

\((6777,6051,7141)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{41}{55}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{142}{165}\right)\)\(e\left(\frac{89}{165}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{73}{165}\right)\)\(e\left(\frac{161}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(191,a) \;\) at \(\;a = \) e.g. 2