Properties

Label 6025.784
Modulus $6025$
Conductor $6025$
Order $40$
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(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,7]))
 
pari: [g,chi] = znchar(Mod(784,6025))
 

Basic properties

Modulus: \(6025\)
Conductor: \(6025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
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.ex

\(\chi_{6025}(194,\cdot)\) \(\chi_{6025}(434,\cdot)\) \(\chi_{6025}(784,\cdot)\) \(\chi_{6025}(2044,\cdot)\) \(\chi_{6025}(2164,\cdot)\) \(\chi_{6025}(2294,\cdot)\) \(\chi_{6025}(2489,\cdot)\) \(\chi_{6025}(3829,\cdot)\) \(\chi_{6025}(4144,\cdot)\) \(\chi_{6025}(4379,\cdot)\) \(\chi_{6025}(4759,\cdot)\) \(\chi_{6025}(4779,\cdot)\) \(\chi_{6025}(5109,\cdot)\) \(\chi_{6025}(5329,\cdot)\) \(\chi_{6025}(5464,\cdot)\) \(\chi_{6025}(5789,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((2652,2176)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{7}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6025 }(784, a) \) \(1\)\(1\)\(e\left(\frac{19}{20}\right)\)\(-i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{17}{20}\right)\)\(-1\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{21}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6025 }(784,a) \;\) at \(\;a = \) e.g. 2