Properties

Label 7623.1508
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([55,11,6]))
 
pari: [g,chi] = znchar(Mod(1508,7623))
 

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.dt

\(\chi_{7623}(551,\cdot)\) \(\chi_{7623}(815,\cdot)\) \(\chi_{7623}(1244,\cdot)\) \(\chi_{7623}(1508,\cdot)\) \(\chi_{7623}(2201,\cdot)\) \(\chi_{7623}(2630,\cdot)\) \(\chi_{7623}(2894,\cdot)\) \(\chi_{7623}(3323,\cdot)\) \(\chi_{7623}(3587,\cdot)\) \(\chi_{7623}(4016,\cdot)\) \(\chi_{7623}(4280,\cdot)\) \(\chi_{7623}(4709,\cdot)\) \(\chi_{7623}(4973,\cdot)\) \(\chi_{7623}(5402,\cdot)\) \(\chi_{7623}(5666,\cdot)\) \(\chi_{7623}(6095,\cdot)\) \(\chi_{7623}(6359,\cdot)\) \(\chi_{7623}(6788,\cdot)\) \(\chi_{7623}(7052,\cdot)\) \(\chi_{7623}(7481,\cdot)\)

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

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

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1508, a) \) \(1\)\(1\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{8}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1508,a) \;\) at \(\;a = \) e.g. 2