Properties

Label 3600.109
Modulus $3600$
Conductor $400$
Order $20$
Real no
Primitive no
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(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,0,14]))
 
pari: [g,chi] = znchar(Mod(109,3600))
 

Basic properties

Modulus: \(3600\)
Conductor: \(400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{400}(109,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3600.ep

\(\chi_{3600}(109,\cdot)\) \(\chi_{3600}(469,\cdot)\) \(\chi_{3600}(829,\cdot)\) \(\chi_{3600}(1189,\cdot)\) \(\chi_{3600}(1909,\cdot)\) \(\chi_{3600}(2269,\cdot)\) \(\chi_{3600}(2629,\cdot)\) \(\chi_{3600}(2989,\cdot)\)

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: 20.20.20971520000000000000000000000000000000000.1

Values on generators

\((3151,901,2801,577)\) → \((1,-i,1,e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3600 }(109, a) \) \(1\)\(1\)\(1\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3600 }(109,a) \;\) at \(\;a = \) e.g. 2