Properties

Label 3200.79
Modulus $3200$
Conductor $800$
Order $40$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,25,4]))
 
pari: [g,chi] = znchar(Mod(79,3200))
 

Basic properties

Modulus: \(3200\)
Conductor: \(800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{800}(779,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3200.cp

\(\chi_{3200}(79,\cdot)\) \(\chi_{3200}(239,\cdot)\) \(\chi_{3200}(559,\cdot)\) \(\chi_{3200}(719,\cdot)\) \(\chi_{3200}(879,\cdot)\) \(\chi_{3200}(1039,\cdot)\) \(\chi_{3200}(1359,\cdot)\) \(\chi_{3200}(1519,\cdot)\) \(\chi_{3200}(1679,\cdot)\) \(\chi_{3200}(1839,\cdot)\) \(\chi_{3200}(2159,\cdot)\) \(\chi_{3200}(2319,\cdot)\) \(\chi_{3200}(2479,\cdot)\) \(\chi_{3200}(2639,\cdot)\) \(\chi_{3200}(2959,\cdot)\) \(\chi_{3200}(3119,\cdot)\)

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: 40.0.15474250491067253436239052800000000000000000000000000000000000000000000000000000000000000000000.1

Values on generators

\((1151,901,2177)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 3200 }(79, a) \) \(-1\)\(1\)\(e\left(\frac{3}{40}\right)\)\(i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{9}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3200 }(79,a) \;\) at \(\;a = \) e.g. 2