Properties

Label 3600.61
Modulus $3600$
Conductor $3600$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,45,40,48]))
 
pari: [g,chi] = znchar(Mod(61,3600))
 

Basic properties

Modulus: \(3600\)
Conductor: \(3600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 3600.ga

\(\chi_{3600}(61,\cdot)\) \(\chi_{3600}(421,\cdot)\) \(\chi_{3600}(661,\cdot)\) \(\chi_{3600}(781,\cdot)\) \(\chi_{3600}(1021,\cdot)\) \(\chi_{3600}(1141,\cdot)\) \(\chi_{3600}(1381,\cdot)\) \(\chi_{3600}(1741,\cdot)\) \(\chi_{3600}(1861,\cdot)\) \(\chi_{3600}(2221,\cdot)\) \(\chi_{3600}(2461,\cdot)\) \(\chi_{3600}(2581,\cdot)\) \(\chi_{3600}(2821,\cdot)\) \(\chi_{3600}(2941,\cdot)\) \(\chi_{3600}(3181,\cdot)\) \(\chi_{3600}(3541,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((3151,901,2801,577)\) → \((1,-i,e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3600 }(61, a) \) \(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3600 }(61,a) \;\) at \(\;a = \) e.g. 2