Properties

Label 8470.1641
Modulus $8470$
Conductor $847$
Order $330$
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([0,55,153]))
 
pari: [g,chi] = znchar(Mod(1641,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(794,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.dj

\(\chi_{8470}(61,\cdot)\) \(\chi_{8470}(101,\cdot)\) \(\chi_{8470}(171,\cdot)\) \(\chi_{8470}(271,\cdot)\) \(\chi_{8470}(381,\cdot)\) \(\chi_{8470}(591,\cdot)\) \(\chi_{8470}(761,\cdot)\) \(\chi_{8470}(831,\cdot)\) \(\chi_{8470}(871,\cdot)\) \(\chi_{8470}(1041,\cdot)\) \(\chi_{8470}(1151,\cdot)\) \(\chi_{8470}(1251,\cdot)\) \(\chi_{8470}(1361,\cdot)\) \(\chi_{8470}(1531,\cdot)\) \(\chi_{8470}(1601,\cdot)\) \(\chi_{8470}(1641,\cdot)\) \(\chi_{8470}(1711,\cdot)\) \(\chi_{8470}(1811,\cdot)\) \(\chi_{8470}(1921,\cdot)\) \(\chi_{8470}(2021,\cdot)\) \(\chi_{8470}(2131,\cdot)\) \(\chi_{8470}(2301,\cdot)\) \(\chi_{8470}(2371,\cdot)\) \(\chi_{8470}(2481,\cdot)\) \(\chi_{8470}(2691,\cdot)\) \(\chi_{8470}(2791,\cdot)\) \(\chi_{8470}(3071,\cdot)\) \(\chi_{8470}(3141,\cdot)\) \(\chi_{8470}(3181,\cdot)\) \(\chi_{8470}(3251,\cdot)\) ...

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 330 polynomial (not computed)

Values on generators

\((6777,6051,7141)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{51}{110}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{18}{55}\right)\)\(e\left(\frac{146}{165}\right)\)\(e\left(\frac{52}{165}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{97}{110}\right)\)\(e\left(\frac{13}{330}\right)\)\(e\left(\frac{133}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(1641,a) \;\) at \(\;a = \) e.g. 2