Properties

Label 7623.2195
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([165,220,147]))
 
pari: [g,chi] = znchar(Mod(2195,7623))
 

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 \(\chi_{2541}(2195,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.fk

\(\chi_{7623}(107,\cdot)\) \(\chi_{7623}(116,\cdot)\) \(\chi_{7623}(305,\cdot)\) \(\chi_{7623}(359,\cdot)\) \(\chi_{7623}(431,\cdot)\) \(\chi_{7623}(557,\cdot)\) \(\chi_{7623}(611,\cdot)\) \(\chi_{7623}(800,\cdot)\) \(\chi_{7623}(809,\cdot)\) \(\chi_{7623}(926,\cdot)\) \(\chi_{7623}(998,\cdot)\) \(\chi_{7623}(1052,\cdot)\) \(\chi_{7623}(1124,\cdot)\) \(\chi_{7623}(1493,\cdot)\) \(\chi_{7623}(1502,\cdot)\) \(\chi_{7623}(1619,\cdot)\) \(\chi_{7623}(1745,\cdot)\) \(\chi_{7623}(1817,\cdot)\) \(\chi_{7623}(1943,\cdot)\) \(\chi_{7623}(1997,\cdot)\) \(\chi_{7623}(2186,\cdot)\) \(\chi_{7623}(2195,\cdot)\) \(\chi_{7623}(2312,\cdot)\) \(\chi_{7623}(2384,\cdot)\) \(\chi_{7623}(2438,\cdot)\) \(\chi_{7623}(2510,\cdot)\) \(\chi_{7623}(2636,\cdot)\) \(\chi_{7623}(2690,\cdot)\) \(\chi_{7623}(2879,\cdot)\) \(\chi_{7623}(2888,\cdot)\) ...

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

\((848,4357,4600)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{49}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(2195, a) \) \(1\)\(1\)\(e\left(\frac{46}{165}\right)\)\(e\left(\frac{92}{165}\right)\)\(e\left(\frac{263}{330}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{109}{110}\right)\)\(e\left(\frac{19}{165}\right)\)\(e\left(\frac{164}{165}\right)\)\(e\left(\frac{101}{330}\right)\)\(e\left(\frac{39}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(2195,a) \;\) at \(\;a = \) e.g. 2