Properties

Label 6025.531
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([48,1]))
 
pari: [g,chi] = znchar(Mod(531,6025))
 

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.ij

\(\chi_{6025}(191,\cdot)\) \(\chi_{6025}(221,\cdot)\) \(\chi_{6025}(291,\cdot)\) \(\chi_{6025}(321,\cdot)\) \(\chi_{6025}(511,\cdot)\) \(\chi_{6025}(531,\cdot)\) \(\chi_{6025}(911,\cdot)\) \(\chi_{6025}(1036,\cdot)\) \(\chi_{6025}(1041,\cdot)\) \(\chi_{6025}(1146,\cdot)\) \(\chi_{6025}(1156,\cdot)\) \(\chi_{6025}(1371,\cdot)\) \(\chi_{6025}(1521,\cdot)\) \(\chi_{6025}(1746,\cdot)\) \(\chi_{6025}(1931,\cdot)\) \(\chi_{6025}(2181,\cdot)\) \(\chi_{6025}(2571,\cdot)\) \(\chi_{6025}(2606,\cdot)\) \(\chi_{6025}(2671,\cdot)\) \(\chi_{6025}(3061,\cdot)\) \(\chi_{6025}(3066,\cdot)\) \(\chi_{6025}(3186,\cdot)\) \(\chi_{6025}(3241,\cdot)\) \(\chi_{6025}(3266,\cdot)\) \(\chi_{6025}(3441,\cdot)\) \(\chi_{6025}(3586,\cdot)\) \(\chi_{6025}(4561,\cdot)\) \(\chi_{6025}(5106,\cdot)\) \(\chi_{6025}(5466,\cdot)\) \(\chi_{6025}(5531,\cdot)\) ...

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

Values on generators

\((2652,2176)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{1}{120}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6025 }(531, a) \) \(1\)\(1\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{120}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{73}{120}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{119}{120}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6025 }(531,a) \;\) at \(\;a = \) e.g. 2