Properties

Label 7200.71
Modulus $7200$
Conductor $1200$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,5,10,12]))
 
pari: [g,chi] = znchar(Mod(71,7200))
 

Basic properties

Modulus: \(7200\)
Conductor: \(1200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1200}(971,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7200.fg

\(\chi_{7200}(71,\cdot)\) \(\chi_{7200}(791,\cdot)\) \(\chi_{7200}(1511,\cdot)\) \(\chi_{7200}(2231,\cdot)\) \(\chi_{7200}(3671,\cdot)\) \(\chi_{7200}(4391,\cdot)\) \(\chi_{7200}(5111,\cdot)\) \(\chi_{7200}(5831,\cdot)\)

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: Number field defined by a degree 20 polynomial

Values on generators

\((6751,901,6401,577)\) → \((-1,i,-1,e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 7200 }(71, a) \) \(1\)\(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{2}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7200 }(71,a) \;\) at \(\;a = \) e.g. 2