Properties

Label 8470.4517
Modulus $8470$
Conductor $385$
Order $60$
Real no
Primitive no
Minimal no
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(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,20,42]))
 
pari: [g,chi] = znchar(Mod(4517,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(385\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{385}(282,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.ci

\(\chi_{8470}(233,\cdot)\) \(\chi_{8470}(403,\cdot)\) \(\chi_{8470}(457,\cdot)\) \(\chi_{8470}(723,\cdot)\) \(\chi_{8470}(1927,\cdot)\) \(\chi_{8470}(2097,\cdot)\) \(\chi_{8470}(2417,\cdot)\) \(\chi_{8470}(2823,\cdot)\) \(\chi_{8470}(4517,\cdot)\) \(\chi_{8470}(4813,\cdot)\) \(\chi_{8470}(6283,\cdot)\) \(\chi_{8470}(6507,\cdot)\) \(\chi_{8470}(6773,\cdot)\) \(\chi_{8470}(7233,\cdot)\) \(\chi_{8470}(7977,\cdot)\) \(\chi_{8470}(8467,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((6777,6051,7141)\) → \((i,e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{19}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(4517,a) \;\) at \(\;a = \) e.g. 2