Properties

Label 7623.3200
Modulus $7623$
Conductor $1089$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,0,27]))
 
pari: [g,chi] = znchar(Mod(3200,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(1089\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1089}(1022,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.eh

\(\chi_{7623}(428,\cdot)\) \(\chi_{7623}(659,\cdot)\) \(\chi_{7623}(1121,\cdot)\) \(\chi_{7623}(1352,\cdot)\) \(\chi_{7623}(2045,\cdot)\) \(\chi_{7623}(2507,\cdot)\) \(\chi_{7623}(2738,\cdot)\) \(\chi_{7623}(3200,\cdot)\) \(\chi_{7623}(3431,\cdot)\) \(\chi_{7623}(3893,\cdot)\) \(\chi_{7623}(4124,\cdot)\) \(\chi_{7623}(4586,\cdot)\) \(\chi_{7623}(4817,\cdot)\) \(\chi_{7623}(5279,\cdot)\) \(\chi_{7623}(5510,\cdot)\) \(\chi_{7623}(5972,\cdot)\) \(\chi_{7623}(6203,\cdot)\) \(\chi_{7623}(6665,\cdot)\) \(\chi_{7623}(7358,\cdot)\) \(\chi_{7623}(7589,\cdot)\)

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

\((848,4357,4600)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{9}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(3200, a) \) \(1\)\(1\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{61}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(3200,a) \;\) at \(\;a = \) e.g. 2