Properties

Label 7623.536
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([275,220,309]))
 
pari: [g,chi] = znchar(Mod(536,7623))
 

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.fg

\(\chi_{7623}(2,\cdot)\) \(\chi_{7623}(95,\cdot)\) \(\chi_{7623}(128,\cdot)\) \(\chi_{7623}(347,\cdot)\) \(\chi_{7623}(380,\cdot)\) \(\chi_{7623}(536,\cdot)\) \(\chi_{7623}(569,\cdot)\) \(\chi_{7623}(662,\cdot)\) \(\chi_{7623}(695,\cdot)\) \(\chi_{7623}(788,\cdot)\) \(\chi_{7623}(821,\cdot)\) \(\chi_{7623}(1040,\cdot)\) \(\chi_{7623}(1073,\cdot)\) \(\chi_{7623}(1229,\cdot)\) \(\chi_{7623}(1262,\cdot)\) \(\chi_{7623}(1355,\cdot)\) \(\chi_{7623}(1388,\cdot)\) \(\chi_{7623}(1481,\cdot)\) \(\chi_{7623}(1514,\cdot)\) \(\chi_{7623}(1733,\cdot)\) \(\chi_{7623}(1766,\cdot)\) \(\chi_{7623}(1922,\cdot)\) \(\chi_{7623}(1955,\cdot)\) \(\chi_{7623}(2081,\cdot)\) \(\chi_{7623}(2174,\cdot)\) \(\chi_{7623}(2207,\cdot)\) \(\chi_{7623}(2426,\cdot)\) \(\chi_{7623}(2459,\cdot)\) \(\chi_{7623}(2615,\cdot)\) \(\chi_{7623}(2648,\cdot)\) ...

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

\((848,4357,4600)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{3}\right),e\left(\frac{103}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(536, a) \) \(1\)\(1\)\(e\left(\frac{17}{165}\right)\)\(e\left(\frac{34}{165}\right)\)\(e\left(\frac{87}{110}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{79}{330}\right)\)\(e\left(\frac{68}{165}\right)\)\(e\left(\frac{8}{165}\right)\)\(e\left(\frac{17}{330}\right)\)\(e\left(\frac{329}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(536,a) \;\) at \(\;a = \) e.g. 2