Properties

Label 6300.13
Modulus $6300$
Conductor $1575$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6300, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,20,57,30]))
 
pari: [g,chi] = znchar(Mod(13,6300))
 

Basic properties

Modulus: \(6300\)
Conductor: \(1575\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1575}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6300.iq

\(\chi_{6300}(13,\cdot)\) \(\chi_{6300}(97,\cdot)\) \(\chi_{6300}(517,\cdot)\) \(\chi_{6300}(853,\cdot)\) \(\chi_{6300}(1273,\cdot)\) \(\chi_{6300}(1777,\cdot)\) \(\chi_{6300}(2113,\cdot)\) \(\chi_{6300}(2533,\cdot)\) \(\chi_{6300}(2617,\cdot)\) \(\chi_{6300}(3037,\cdot)\) \(\chi_{6300}(3373,\cdot)\) \(\chi_{6300}(3877,\cdot)\) \(\chi_{6300}(4297,\cdot)\) \(\chi_{6300}(4633,\cdot)\) \(\chi_{6300}(5053,\cdot)\) \(\chi_{6300}(5137,\cdot)\)

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

Values on generators

\((3151,2801,3277,3601)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{19}{20}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 6300 }(13, a) \) \(1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6300 }(13,a) \;\) at \(\;a = \) e.g. 2