Properties

Label 7623.5785
Modulus $7623$
Conductor $7623$
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(7623, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,11,9]))
 
pari: [g,chi] = znchar(Mod(5785,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(7623\)
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 7623.eo

\(\chi_{7623}(670,\cdot)\) \(\chi_{7623}(934,\cdot)\) \(\chi_{7623}(1363,\cdot)\) \(\chi_{7623}(1627,\cdot)\) \(\chi_{7623}(2320,\cdot)\) \(\chi_{7623}(2749,\cdot)\) \(\chi_{7623}(3013,\cdot)\) \(\chi_{7623}(3442,\cdot)\) \(\chi_{7623}(3706,\cdot)\) \(\chi_{7623}(4135,\cdot)\) \(\chi_{7623}(4399,\cdot)\) \(\chi_{7623}(4828,\cdot)\) \(\chi_{7623}(5092,\cdot)\) \(\chi_{7623}(5521,\cdot)\) \(\chi_{7623}(5785,\cdot)\) \(\chi_{7623}(6214,\cdot)\) \(\chi_{7623}(6478,\cdot)\) \(\chi_{7623}(6907,\cdot)\) \(\chi_{7623}(7171,\cdot)\) \(\chi_{7623}(7600,\cdot)\)

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

\((848,4357,4600)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{6}\right),e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(5785, a) \) \(1\)\(1\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{35}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(5785,a) \;\) at \(\;a = \) e.g. 2