Properties

Label 6025.1182
Modulus $6025$
Conductor $1205$
Order $80$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6025, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,59]))
 
pari: [g,chi] = znchar(Mod(1182,6025))
 

Basic properties

Modulus: \(6025\)
Conductor: \(1205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1205}(1182,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6025.ha

\(\chi_{6025}(43,\cdot)\) \(\chi_{6025}(343,\cdot)\) \(\chi_{6025}(943,\cdot)\) \(\chi_{6025}(1057,\cdot)\) \(\chi_{6025}(1132,\cdot)\) \(\chi_{6025}(1182,\cdot)\) \(\chi_{6025}(1582,\cdot)\) \(\chi_{6025}(2068,\cdot)\) \(\chi_{6025}(2293,\cdot)\) \(\chi_{6025}(2307,\cdot)\) \(\chi_{6025}(2382,\cdot)\) \(\chi_{6025}(2393,\cdot)\) \(\chi_{6025}(2443,\cdot)\) \(\chi_{6025}(2618,\cdot)\) \(\chi_{6025}(2668,\cdot)\) \(\chi_{6025}(2768,\cdot)\) \(\chi_{6025}(2807,\cdot)\) \(\chi_{6025}(2993,\cdot)\) \(\chi_{6025}(3107,\cdot)\) \(\chi_{6025}(3882,\cdot)\) \(\chi_{6025}(4118,\cdot)\) \(\chi_{6025}(4182,\cdot)\) \(\chi_{6025}(4607,\cdot)\) \(\chi_{6025}(4682,\cdot)\) \(\chi_{6025}(4718,\cdot)\) \(\chi_{6025}(5018,\cdot)\) \(\chi_{6025}(5118,\cdot)\) \(\chi_{6025}(5407,\cdot)\) \(\chi_{6025}(5807,\cdot)\) \(\chi_{6025}(5857,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((2652,2176)\) → \((i,e\left(\frac{59}{80}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6025 }(1182, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{39}{40}\right)\)\(-i\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{79}{80}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{33}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6025 }(1182,a) \;\) at \(\;a = \) e.g. 2