Properties

Label 7623.89
Modulus $7623$
Conductor $2541$
Order $66$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,55,36]))
 
pari: [g,chi] = znchar(Mod(89,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(2541\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2541}(89,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.dz

\(\chi_{7623}(89,\cdot)\) \(\chi_{7623}(584,\cdot)\) \(\chi_{7623}(782,\cdot)\) \(\chi_{7623}(1277,\cdot)\) \(\chi_{7623}(1475,\cdot)\) \(\chi_{7623}(1970,\cdot)\) \(\chi_{7623}(2168,\cdot)\) \(\chi_{7623}(2861,\cdot)\) \(\chi_{7623}(3356,\cdot)\) \(\chi_{7623}(3554,\cdot)\) \(\chi_{7623}(4049,\cdot)\) \(\chi_{7623}(4247,\cdot)\) \(\chi_{7623}(4742,\cdot)\) \(\chi_{7623}(4940,\cdot)\) \(\chi_{7623}(5435,\cdot)\) \(\chi_{7623}(5633,\cdot)\) \(\chi_{7623}(6128,\cdot)\) \(\chi_{7623}(6326,\cdot)\) \(\chi_{7623}(6821,\cdot)\) \(\chi_{7623}(7514,\cdot)\)

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

\((848,4357,4600)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(89, a) \) \(1\)\(1\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{5}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(89,a) \;\) at \(\;a = \) e.g. 2