Properties

Label 6025.164
Modulus $6025$
Conductor $6025$
Order $120$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6025, base_ring=CyclotomicField(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([36,73]))
 
pari: [g,chi] = znchar(Mod(164,6025))
 

Basic properties

Modulus: \(6025\)
Conductor: \(6025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6025.ih

\(\chi_{6025}(164,\cdot)\) \(\chi_{6025}(229,\cdot)\) \(\chi_{6025}(259,\cdot)\) \(\chi_{6025}(479,\cdot)\) \(\chi_{6025}(914,\cdot)\) \(\chi_{6025}(944,\cdot)\) \(\chi_{6025}(1014,\cdot)\) \(\chi_{6025}(1044,\cdot)\) \(\chi_{6025}(1234,\cdot)\) \(\chi_{6025}(1254,\cdot)\) \(\chi_{6025}(1634,\cdot)\) \(\chi_{6025}(1759,\cdot)\) \(\chi_{6025}(1764,\cdot)\) \(\chi_{6025}(1869,\cdot)\) \(\chi_{6025}(1879,\cdot)\) \(\chi_{6025}(2094,\cdot)\) \(\chi_{6025}(2244,\cdot)\) \(\chi_{6025}(2469,\cdot)\) \(\chi_{6025}(2654,\cdot)\) \(\chi_{6025}(2904,\cdot)\) \(\chi_{6025}(3294,\cdot)\) \(\chi_{6025}(3329,\cdot)\) \(\chi_{6025}(3394,\cdot)\) \(\chi_{6025}(3784,\cdot)\) \(\chi_{6025}(3789,\cdot)\) \(\chi_{6025}(3909,\cdot)\) \(\chi_{6025}(3964,\cdot)\) \(\chi_{6025}(3989,\cdot)\) \(\chi_{6025}(4164,\cdot)\) \(\chi_{6025}(4309,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((2652,2176)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{73}{120}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6025 }(164, a) \) \(1\)\(1\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{13}{120}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{120}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6025 }(164,a) \;\) at \(\;a = \) e.g. 2