Properties

Label 7623.1664
Modulus $7623$
Conductor $2541$
Order $330$
Real no
Primitive no
Minimal no
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([165,275,324]))
 
pari: [g,chi] = znchar(Mod(1664,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(2541\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2541}(1664,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.fr

\(\chi_{7623}(26,\cdot)\) \(\chi_{7623}(80,\cdot)\) \(\chi_{7623}(152,\cdot)\) \(\chi_{7623}(278,\cdot)\) \(\chi_{7623}(467,\cdot)\) \(\chi_{7623}(521,\cdot)\) \(\chi_{7623}(647,\cdot)\) \(\chi_{7623}(719,\cdot)\) \(\chi_{7623}(773,\cdot)\) \(\chi_{7623}(845,\cdot)\) \(\chi_{7623}(962,\cdot)\) \(\chi_{7623}(1160,\cdot)\) \(\chi_{7623}(1214,\cdot)\) \(\chi_{7623}(1466,\cdot)\) \(\chi_{7623}(1538,\cdot)\) \(\chi_{7623}(1655,\cdot)\) \(\chi_{7623}(1664,\cdot)\) \(\chi_{7623}(1853,\cdot)\) \(\chi_{7623}(1907,\cdot)\) \(\chi_{7623}(2033,\cdot)\) \(\chi_{7623}(2105,\cdot)\) \(\chi_{7623}(2159,\cdot)\) \(\chi_{7623}(2231,\cdot)\) \(\chi_{7623}(2348,\cdot)\) \(\chi_{7623}(2357,\cdot)\) \(\chi_{7623}(2546,\cdot)\) \(\chi_{7623}(2600,\cdot)\) \(\chi_{7623}(2726,\cdot)\) \(\chi_{7623}(2798,\cdot)\) \(\chi_{7623}(2852,\cdot)\) ...

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

\((848,4357,4600)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{54}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1664, a) \) \(1\)\(1\)\(e\left(\frac{49}{330}\right)\)\(e\left(\frac{49}{165}\right)\)\(e\left(\frac{53}{165}\right)\)\(e\left(\frac{49}{110}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{73}{110}\right)\)\(e\left(\frac{98}{165}\right)\)\(e\left(\frac{73}{165}\right)\)\(e\left(\frac{217}{330}\right)\)\(e\left(\frac{34}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1664,a) \;\) at \(\;a = \) e.g. 2