Properties

Label 6025.1907
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([20,21]))
 
pari: [g,chi] = znchar(Mod(1907,6025))
 

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 \(\chi_{1205}(702,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6025.hb

\(\chi_{6025}(57,\cdot)\) \(\chi_{6025}(93,\cdot)\) \(\chi_{6025}(168,\cdot)\) \(\chi_{6025}(218,\cdot)\) \(\chi_{6025}(618,\cdot)\) \(\chi_{6025}(907,\cdot)\) \(\chi_{6025}(1007,\cdot)\) \(\chi_{6025}(1307,\cdot)\) \(\chi_{6025}(1343,\cdot)\) \(\chi_{6025}(1418,\cdot)\) \(\chi_{6025}(1843,\cdot)\) \(\chi_{6025}(1907,\cdot)\) \(\chi_{6025}(2143,\cdot)\) \(\chi_{6025}(2918,\cdot)\) \(\chi_{6025}(3032,\cdot)\) \(\chi_{6025}(3218,\cdot)\) \(\chi_{6025}(3257,\cdot)\) \(\chi_{6025}(3357,\cdot)\) \(\chi_{6025}(3407,\cdot)\) \(\chi_{6025}(3582,\cdot)\) \(\chi_{6025}(3632,\cdot)\) \(\chi_{6025}(3643,\cdot)\) \(\chi_{6025}(3718,\cdot)\) \(\chi_{6025}(3732,\cdot)\) \(\chi_{6025}(3957,\cdot)\) \(\chi_{6025}(4443,\cdot)\) \(\chi_{6025}(4843,\cdot)\) \(\chi_{6025}(4893,\cdot)\) \(\chi_{6025}(4968,\cdot)\) \(\chi_{6025}(5082,\cdot)\) ...

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

\((2652,2176)\) → \((i,e\left(\frac{21}{80}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6025 }(1907, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{21}{40}\right)\)\(i\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{41}{80}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{7}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6025 }(1907,a) \;\) at \(\;a = \) e.g. 2