Properties

Label 8470.139
Modulus $8470$
Conductor $4235$
Order $110$
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([55,55,67]))
 
pari: [g,chi] = znchar(Mod(139,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(139,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cs

\(\chi_{8470}(139,\cdot)\) \(\chi_{8470}(349,\cdot)\) \(\chi_{8470}(629,\cdot)\) \(\chi_{8470}(909,\cdot)\) \(\chi_{8470}(1119,\cdot)\) \(\chi_{8470}(1399,\cdot)\) \(\chi_{8470}(1469,\cdot)\) \(\chi_{8470}(1679,\cdot)\) \(\chi_{8470}(1889,\cdot)\) \(\chi_{8470}(2239,\cdot)\) \(\chi_{8470}(2449,\cdot)\) \(\chi_{8470}(2939,\cdot)\) \(\chi_{8470}(3009,\cdot)\) \(\chi_{8470}(3219,\cdot)\) \(\chi_{8470}(3429,\cdot)\) \(\chi_{8470}(3709,\cdot)\) \(\chi_{8470}(3779,\cdot)\) \(\chi_{8470}(3989,\cdot)\) \(\chi_{8470}(4199,\cdot)\) \(\chi_{8470}(4479,\cdot)\) \(\chi_{8470}(4549,\cdot)\) \(\chi_{8470}(4969,\cdot)\) \(\chi_{8470}(5249,\cdot)\) \(\chi_{8470}(5319,\cdot)\) \(\chi_{8470}(5529,\cdot)\) \(\chi_{8470}(5739,\cdot)\) \(\chi_{8470}(6019,\cdot)\) \(\chi_{8470}(6089,\cdot)\) \(\chi_{8470}(6299,\cdot)\) \(\chi_{8470}(6509,\cdot)\) ...

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 110 polynomial (not computed)

Values on generators

\((6777,6051,7141)\) → \((-1,-1,e\left(\frac{67}{110}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{57}{110}\right)\)\(e\left(\frac{93}{110}\right)\)\(e\left(\frac{3}{55}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{97}{110}\right)\)\(e\left(\frac{9}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(139,a) \;\) at \(\;a = \) e.g. 2