Properties

Label 7200.847
Modulus $7200$
Conductor $200$
Order $20$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,10,0,17]))
 
pari: [g,chi] = znchar(Mod(847,7200))
 

Basic properties

Modulus: \(7200\)
Conductor: \(200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{200}(147,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7200.ep

\(\chi_{7200}(847,\cdot)\) \(\chi_{7200}(1423,\cdot)\) \(\chi_{7200}(2287,\cdot)\) \(\chi_{7200}(2863,\cdot)\) \(\chi_{7200}(3727,\cdot)\) \(\chi_{7200}(4303,\cdot)\) \(\chi_{7200}(5167,\cdot)\) \(\chi_{7200}(7183,\cdot)\)

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: 20.20.3125000000000000000000000000000000.1

Values on generators

\((6751,901,6401,577)\) → \((-1,-1,1,e\left(\frac{17}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 7200 }(847, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{2}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7200 }(847,a) \;\) at \(\;a = \) e.g. 2