Properties

Label 3600.1583
Modulus $3600$
Conductor $300$
Order $20$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,0,10,3]))
 
pari: [g,chi] = znchar(Mod(1583,3600))
 

Basic properties

Modulus: \(3600\)
Conductor: \(300\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{300}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3600.dx

\(\chi_{3600}(287,\cdot)\) \(\chi_{3600}(863,\cdot)\) \(\chi_{3600}(1583,\cdot)\) \(\chi_{3600}(1727,\cdot)\) \(\chi_{3600}(2303,\cdot)\) \(\chi_{3600}(2447,\cdot)\) \(\chi_{3600}(3023,\cdot)\) \(\chi_{3600}(3167,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.0.180203247070312500000000000000000000.1

Values on generators

\((3151,901,2801,577)\) → \((-1,1,-1,e\left(\frac{3}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3600 }(1583, a) \) \(-1\)\(1\)\(i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3600 }(1583,a) \;\) at \(\;a = \) e.g. 2