Properties

Label 7623.1643
Modulus $7623$
Conductor $7623$
Order $330$
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(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([275,275,246]))
 
pari: [g,chi] = znchar(Mod(1643,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(7623\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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.fc

\(\chi_{7623}(5,\cdot)\) \(\chi_{7623}(38,\cdot)\) \(\chi_{7623}(257,\cdot)\) \(\chi_{7623}(290,\cdot)\) \(\chi_{7623}(383,\cdot)\) \(\chi_{7623}(416,\cdot)\) \(\chi_{7623}(509,\cdot)\) \(\chi_{7623}(542,\cdot)\) \(\chi_{7623}(698,\cdot)\) \(\chi_{7623}(731,\cdot)\) \(\chi_{7623}(950,\cdot)\) \(\chi_{7623}(983,\cdot)\) \(\chi_{7623}(1076,\cdot)\) \(\chi_{7623}(1109,\cdot)\) \(\chi_{7623}(1202,\cdot)\) \(\chi_{7623}(1235,\cdot)\) \(\chi_{7623}(1391,\cdot)\) \(\chi_{7623}(1424,\cdot)\) \(\chi_{7623}(1643,\cdot)\) \(\chi_{7623}(1676,\cdot)\) \(\chi_{7623}(1769,\cdot)\) \(\chi_{7623}(1802,\cdot)\) \(\chi_{7623}(1895,\cdot)\) \(\chi_{7623}(1928,\cdot)\) \(\chi_{7623}(2117,\cdot)\) \(\chi_{7623}(2336,\cdot)\) \(\chi_{7623}(2369,\cdot)\) \(\chi_{7623}(2462,\cdot)\) \(\chi_{7623}(2495,\cdot)\) \(\chi_{7623}(2588,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((848,4357,4600)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{5}{6}\right),e\left(\frac{41}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1643, a) \) \(1\)\(1\)\(e\left(\frac{27}{110}\right)\)\(e\left(\frac{27}{55}\right)\)\(e\left(\frac{82}{165}\right)\)\(e\left(\frac{81}{110}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{151}{330}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{142}{165}\right)\)\(e\left(\frac{13}{330}\right)\)\(e\left(\frac{163}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1643,a) \;\) at \(\;a = \) e.g. 2