Properties

Label 4800.11
Modulus $4800$
Conductor $4800$
Order $80$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4800, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,25,40,64]))
 
pari: [g,chi] = znchar(Mod(11,4800))
 

Basic properties

Modulus: \(4800\)
Conductor: \(4800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
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 4800.fe

\(\chi_{4800}(11,\cdot)\) \(\chi_{4800}(131,\cdot)\) \(\chi_{4800}(371,\cdot)\) \(\chi_{4800}(491,\cdot)\) \(\chi_{4800}(611,\cdot)\) \(\chi_{4800}(731,\cdot)\) \(\chi_{4800}(971,\cdot)\) \(\chi_{4800}(1091,\cdot)\) \(\chi_{4800}(1211,\cdot)\) \(\chi_{4800}(1331,\cdot)\) \(\chi_{4800}(1571,\cdot)\) \(\chi_{4800}(1691,\cdot)\) \(\chi_{4800}(1811,\cdot)\) \(\chi_{4800}(1931,\cdot)\) \(\chi_{4800}(2171,\cdot)\) \(\chi_{4800}(2291,\cdot)\) \(\chi_{4800}(2411,\cdot)\) \(\chi_{4800}(2531,\cdot)\) \(\chi_{4800}(2771,\cdot)\) \(\chi_{4800}(2891,\cdot)\) \(\chi_{4800}(3011,\cdot)\) \(\chi_{4800}(3131,\cdot)\) \(\chi_{4800}(3371,\cdot)\) \(\chi_{4800}(3491,\cdot)\) \(\chi_{4800}(3611,\cdot)\) \(\chi_{4800}(3731,\cdot)\) \(\chi_{4800}(3971,\cdot)\) \(\chi_{4800}(4091,\cdot)\) \(\chi_{4800}(4211,\cdot)\) \(\chi_{4800}(4331,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((4351,901,1601,577)\) → \((-1,e\left(\frac{5}{16}\right),-1,e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4800 }(11, a) \) \(1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{29}{80}\right)\)\(e\left(\frac{71}{80}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{7}{80}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{43}{80}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{80}\right)\)\(e\left(\frac{3}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4800 }(11,a) \;\) at \(\;a = \) e.g. 2