Properties

Label 8470.5799
Modulus $8470$
Conductor $385$
Order $30$
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([15,5,3]))
 
pari: [g,chi] = znchar(Mod(5799,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(385\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{385}(24,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.bz

\(\chi_{8470}(1909,\cdot)\) \(\chi_{8470}(3379,\cdot)\) \(\chi_{8470}(3869,\cdot)\) \(\chi_{8470}(4329,\cdot)\) \(\chi_{8470}(5799,\cdot)\) \(\chi_{8470}(5969,\cdot)\) \(\chi_{8470}(6289,\cdot)\) \(\chi_{8470}(8389,\cdot)\)

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: 30.30.536541803411786167873866979265550759408678479595855712890625.1

Values on generators

\((6777,6051,7141)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{10}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(5799,a) \;\) at \(\;a = \) e.g. 2