Properties

Label 8470.2501
Modulus $8470$
Conductor $77$
Order $15$
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(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10,6]))
 
pari: [g,chi] = znchar(Mod(2501,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.bh

\(\chi_{8470}(81,\cdot)\) \(\chi_{8470}(2181,\cdot)\) \(\chi_{8470}(2501,\cdot)\) \(\chi_{8470}(2671,\cdot)\) \(\chi_{8470}(4141,\cdot)\) \(\chi_{8470}(4601,\cdot)\) \(\chi_{8470}(5091,\cdot)\) \(\chi_{8470}(6561,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 15.15.886528337182930278529.1

Values on generators

\((6777,6051,7141)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{1}{5}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(2501,a) \;\) at \(\;a = \) e.g. 2