Properties

Label 7623.1594
Modulus $7623$
Conductor $847$
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(7623, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,55,57]))
 
pari: [g,chi] = znchar(Mod(1594,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(747,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.dx

\(\chi_{7623}(10,\cdot)\) \(\chi_{7623}(208,\cdot)\) \(\chi_{7623}(703,\cdot)\) \(\chi_{7623}(901,\cdot)\) \(\chi_{7623}(1396,\cdot)\) \(\chi_{7623}(1594,\cdot)\) \(\chi_{7623}(2089,\cdot)\) \(\chi_{7623}(2287,\cdot)\) \(\chi_{7623}(2980,\cdot)\) \(\chi_{7623}(3475,\cdot)\) \(\chi_{7623}(3673,\cdot)\) \(\chi_{7623}(4168,\cdot)\) \(\chi_{7623}(4366,\cdot)\) \(\chi_{7623}(4861,\cdot)\) \(\chi_{7623}(5059,\cdot)\) \(\chi_{7623}(5554,\cdot)\) \(\chi_{7623}(5752,\cdot)\) \(\chi_{7623}(6247,\cdot)\) \(\chi_{7623}(6445,\cdot)\) \(\chi_{7623}(6940,\cdot)\)

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

\((848,4357,4600)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1594, a) \) \(1\)\(1\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{3}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1594,a) \;\) at \(\;a = \) e.g. 2