Properties

Label 6300.121
Modulus $6300$
Conductor $1575$
Order $15$
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([0,10,18,10]))
 
pari: [g,chi] = znchar(Mod(121,6300))
 

Basic properties

Modulus: \(6300\)
Conductor: \(1575\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1575}(121,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6300.fj

\(\chi_{6300}(121,\cdot)\) \(\chi_{6300}(781,\cdot)\) \(\chi_{6300}(1381,\cdot)\) \(\chi_{6300}(2041,\cdot)\) \(\chi_{6300}(2641,\cdot)\) \(\chi_{6300}(4561,\cdot)\) \(\chi_{6300}(5161,\cdot)\) \(\chi_{6300}(5821,\cdot)\)

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 15 polynomial

Values on generators

\((3151,2801,3277,3601)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 6300 }(121, a) \) \(1\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6300 }(121,a) \;\) at \(\;a = \) e.g. 2