Properties

Label 6025.39
Modulus $6025$
Conductor $6025$
Order $240$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6025, base_ring=CyclotomicField(240))
 
M = H._module
 
chi = DirichletCharacter(H, M([72,229]))
 
pari: [g,chi] = znchar(Mod(39,6025))
 

Basic properties

Modulus: \(6025\)
Conductor: \(6025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(240\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6025.jk

\(\chi_{6025}(34,\cdot)\) \(\chi_{6025}(39,\cdot)\) \(\chi_{6025}(204,\cdot)\) \(\chi_{6025}(234,\cdot)\) \(\chi_{6025}(309,\cdot)\) \(\chi_{6025}(469,\cdot)\) \(\chi_{6025}(619,\cdot)\) \(\chi_{6025}(709,\cdot)\) \(\chi_{6025}(769,\cdot)\) \(\chi_{6025}(854,\cdot)\) \(\chi_{6025}(1154,\cdot)\) \(\chi_{6025}(1279,\cdot)\) \(\chi_{6025}(1319,\cdot)\) \(\chi_{6025}(1334,\cdot)\) \(\chi_{6025}(1384,\cdot)\) \(\chi_{6025}(1394,\cdot)\) \(\chi_{6025}(1404,\cdot)\) \(\chi_{6025}(1729,\cdot)\) \(\chi_{6025}(1819,\cdot)\) \(\chi_{6025}(1854,\cdot)\) \(\chi_{6025}(1889,\cdot)\) \(\chi_{6025}(1959,\cdot)\) \(\chi_{6025}(1979,\cdot)\) \(\chi_{6025}(1984,\cdot)\) \(\chi_{6025}(1994,\cdot)\) \(\chi_{6025}(2014,\cdot)\) \(\chi_{6025}(2239,\cdot)\) \(\chi_{6025}(2279,\cdot)\) \(\chi_{6025}(2339,\cdot)\) \(\chi_{6025}(2344,\cdot)\) ...

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: $\Q(\zeta_{240})$
Fixed field: Number field defined by a degree 240 polynomial (not computed)

Values on generators

\((2652,2176)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{229}{240}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6025 }(39, a) \) \(-1\)\(1\)\(e\left(\frac{71}{120}\right)\)\(e\left(\frac{91}{120}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{109}{240}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{157}{240}\right)\)\(e\left(\frac{113}{120}\right)\)\(e\left(\frac{131}{240}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6025 }(39,a) \;\) at \(\;a = \) e.g. 2