Properties

Label 8470.4229
Modulus $8470$
Conductor $605$
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([55,0,34]))
 
pari: [g,chi] = znchar(Mod(4229,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(599,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cv

\(\chi_{8470}(169,\cdot)\) \(\chi_{8470}(379,\cdot)\) \(\chi_{8470}(449,\cdot)\) \(\chi_{8470}(939,\cdot)\) \(\chi_{8470}(1149,\cdot)\) \(\chi_{8470}(1499,\cdot)\) \(\chi_{8470}(1709,\cdot)\) \(\chi_{8470}(1919,\cdot)\) \(\chi_{8470}(1989,\cdot)\) \(\chi_{8470}(2269,\cdot)\) \(\chi_{8470}(2479,\cdot)\) \(\chi_{8470}(2759,\cdot)\) \(\chi_{8470}(3039,\cdot)\) \(\chi_{8470}(3249,\cdot)\) \(\chi_{8470}(3459,\cdot)\) \(\chi_{8470}(3529,\cdot)\) \(\chi_{8470}(3809,\cdot)\) \(\chi_{8470}(4019,\cdot)\) \(\chi_{8470}(4229,\cdot)\) \(\chi_{8470}(4299,\cdot)\) \(\chi_{8470}(4579,\cdot)\) \(\chi_{8470}(4789,\cdot)\) \(\chi_{8470}(4999,\cdot)\) \(\chi_{8470}(5069,\cdot)\) \(\chi_{8470}(5349,\cdot)\) \(\chi_{8470}(5559,\cdot)\) \(\chi_{8470}(5769,\cdot)\) \(\chi_{8470}(5839,\cdot)\) \(\chi_{8470}(6119,\cdot)\) \(\chi_{8470}(6329,\cdot)\) ...

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

\((6777,6051,7141)\) → \((-1,1,e\left(\frac{17}{55}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{79}{110}\right)\)\(e\left(\frac{71}{110}\right)\)\(e\left(\frac{36}{55}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{14}{55}\right)\)\(e\left(\frac{32}{55}\right)\)\(e\left(\frac{53}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(4229,a) \;\) at \(\;a = \) e.g. 2