Properties

Label 4830.641
Modulus $4830$
Conductor $483$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,0,44,15]))
 
pari: [g,chi] = znchar(Mod(641,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(483\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{483}(158,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.de

\(\chi_{4830}(11,\cdot)\) \(\chi_{4830}(191,\cdot)\) \(\chi_{4830}(221,\cdot)\) \(\chi_{4830}(401,\cdot)\) \(\chi_{4830}(431,\cdot)\) \(\chi_{4830}(641,\cdot)\) \(\chi_{4830}(1031,\cdot)\) \(\chi_{4830}(1661,\cdot)\) \(\chi_{4830}(1901,\cdot)\) \(\chi_{4830}(2081,\cdot)\) \(\chi_{4830}(2291,\cdot)\) \(\chi_{4830}(2321,\cdot)\) \(\chi_{4830}(2501,\cdot)\) \(\chi_{4830}(2711,\cdot)\) \(\chi_{4830}(2951,\cdot)\) \(\chi_{4830}(3161,\cdot)\) \(\chi_{4830}(3791,\cdot)\) \(\chi_{4830}(3971,\cdot)\) \(\chi_{4830}(4391,\cdot)\) \(\chi_{4830}(4421,\cdot)\)

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

\((3221,967,2761,1891)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{5}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 4830 }(641, a) \) \(1\)\(1\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{7}{66}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(641,a) \;\) at \(\;a = \) e.g. 2