Properties

Label 5796.83
Modulus $5796$
Conductor $5796$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5796, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,11,33,63]))
 
pari: [g,chi] = znchar(Mod(83,5796))
 

Basic properties

Modulus: \(5796\)
Conductor: \(5796\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 5796.ei

\(\chi_{5796}(83,\cdot)\) \(\chi_{5796}(419,\cdot)\) \(\chi_{5796}(839,\cdot)\) \(\chi_{5796}(1091,\cdot)\) \(\chi_{5796}(1847,\cdot)\) \(\chi_{5796}(2183,\cdot)\) \(\chi_{5796}(2351,\cdot)\) \(\chi_{5796}(2435,\cdot)\) \(\chi_{5796}(2687,\cdot)\) \(\chi_{5796}(3191,\cdot)\) \(\chi_{5796}(3695,\cdot)\) \(\chi_{5796}(3947,\cdot)\) \(\chi_{5796}(4115,\cdot)\) \(\chi_{5796}(4367,\cdot)\) \(\chi_{5796}(4619,\cdot)\) \(\chi_{5796}(4703,\cdot)\) \(\chi_{5796}(4955,\cdot)\) \(\chi_{5796}(5123,\cdot)\) \(\chi_{5796}(5627,\cdot)\) \(\chi_{5796}(5711,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((2899,1289,829,4789)\) → \((-1,e\left(\frac{1}{6}\right),-1,e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5796 }(83, a) \) \(1\)\(1\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{26}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5796 }(83,a) \;\) at \(\;a = \) e.g. 2