Properties

Label 8470.911
Modulus $8470$
Conductor $121$
Order $55$
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(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,6]))
 
pari: [g,chi] = znchar(Mod(911,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(55\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(64,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cf

\(\chi_{8470}(71,\cdot)\) \(\chi_{8470}(141,\cdot)\) \(\chi_{8470}(421,\cdot)\) \(\chi_{8470}(631,\cdot)\) \(\chi_{8470}(841,\cdot)\) \(\chi_{8470}(911,\cdot)\) \(\chi_{8470}(1191,\cdot)\) \(\chi_{8470}(1401,\cdot)\) \(\chi_{8470}(1611,\cdot)\) \(\chi_{8470}(1681,\cdot)\) \(\chi_{8470}(1961,\cdot)\) \(\chi_{8470}(2171,\cdot)\) \(\chi_{8470}(2381,\cdot)\) \(\chi_{8470}(2451,\cdot)\) \(\chi_{8470}(2731,\cdot)\) \(\chi_{8470}(2941,\cdot)\) \(\chi_{8470}(3151,\cdot)\) \(\chi_{8470}(3221,\cdot)\) \(\chi_{8470}(3501,\cdot)\) \(\chi_{8470}(3921,\cdot)\) \(\chi_{8470}(3991,\cdot)\) \(\chi_{8470}(4271,\cdot)\) \(\chi_{8470}(4481,\cdot)\) \(\chi_{8470}(4691,\cdot)\) \(\chi_{8470}(4761,\cdot)\) \(\chi_{8470}(5041,\cdot)\) \(\chi_{8470}(5251,\cdot)\) \(\chi_{8470}(5461,\cdot)\) \(\chi_{8470}(5531,\cdot)\) \(\chi_{8470}(6021,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 55 polynomial

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{28}{55}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{29}{55}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{51}{55}\right)\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{16}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(911,a) \;\) at \(\;a = \) e.g. 2