Properties

Label 7623.1220
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([55,22,45]))
 
pari: [g,chi] = znchar(Mod(1220,7623))
 

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.dq

\(\chi_{7623}(263,\cdot)\) \(\chi_{7623}(527,\cdot)\) \(\chi_{7623}(956,\cdot)\) \(\chi_{7623}(1220,\cdot)\) \(\chi_{7623}(1649,\cdot)\) \(\chi_{7623}(1913,\cdot)\) \(\chi_{7623}(2342,\cdot)\) \(\chi_{7623}(2606,\cdot)\) \(\chi_{7623}(3035,\cdot)\) \(\chi_{7623}(3299,\cdot)\) \(\chi_{7623}(3728,\cdot)\) \(\chi_{7623}(4421,\cdot)\) \(\chi_{7623}(4685,\cdot)\) \(\chi_{7623}(5114,\cdot)\) \(\chi_{7623}(5378,\cdot)\) \(\chi_{7623}(6071,\cdot)\) \(\chi_{7623}(6500,\cdot)\) \(\chi_{7623}(6764,\cdot)\) \(\chi_{7623}(7193,\cdot)\) \(\chi_{7623}(7457,\cdot)\)

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

\((848,4357,4600)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{3}\right),e\left(\frac{15}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1220, a) \) \(1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{43}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1220,a) \;\) at \(\;a = \) e.g. 2