Properties

Label 5166.923
Modulus $5166$
Conductor $2583$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5166, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,30,21]))
 
pari: [g,chi] = znchar(Mod(923,5166))
 

Basic properties

Modulus: \(5166\)
Conductor: \(2583\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2583}(923,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5166.fa

\(\chi_{5166}(923,\cdot)\) \(\chi_{5166}(1427,\cdot)\) \(\chi_{5166}(1553,\cdot)\) \(\chi_{5166}(1679,\cdot)\) \(\chi_{5166}(1847,\cdot)\) \(\chi_{5166}(1973,\cdot)\) \(\chi_{5166}(2099,\cdot)\) \(\chi_{5166}(2603,\cdot)\) \(\chi_{5166}(3569,\cdot)\) \(\chi_{5166}(3695,\cdot)\) \(\chi_{5166}(3821,\cdot)\) \(\chi_{5166}(4325,\cdot)\) \(\chi_{5166}(4367,\cdot)\) \(\chi_{5166}(4871,\cdot)\) \(\chi_{5166}(4997,\cdot)\) \(\chi_{5166}(5123,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((2297,2215,3655)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{7}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 5166 }(923, a) \) \(1\)\(1\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5166 }(923,a) \;\) at \(\;a = \) e.g. 2