Properties

Label 6300.491
Modulus $6300$
Conductor $900$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,25,6,0]))
 
pari: [g,chi] = znchar(Mod(491,6300))
 

Basic properties

Modulus: \(6300\)
Conductor: \(900\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{900}(491,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6300.hn

\(\chi_{6300}(491,\cdot)\) \(\chi_{6300}(911,\cdot)\) \(\chi_{6300}(2171,\cdot)\) \(\chi_{6300}(3011,\cdot)\) \(\chi_{6300}(3431,\cdot)\) \(\chi_{6300}(4271,\cdot)\) \(\chi_{6300}(4691,\cdot)\) \(\chi_{6300}(5531,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((3151,2801,3277,3601)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{1}{5}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 6300 }(491, a) \) \(1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6300 }(491,a) \;\) at \(\;a = \) e.g. 2