Properties

Label 7623.1867
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([110,275,309]))
 
pari: [g,chi] = znchar(Mod(1867,7623))
 

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.fe

\(\chi_{7623}(52,\cdot)\) \(\chi_{7623}(178,\cdot)\) \(\chi_{7623}(292,\cdot)\) \(\chi_{7623}(304,\cdot)\) \(\chi_{7623}(556,\cdot)\) \(\chi_{7623}(607,\cdot)\) \(\chi_{7623}(733,\cdot)\) \(\chi_{7623}(745,\cdot)\) \(\chi_{7623}(871,\cdot)\) \(\chi_{7623}(985,\cdot)\) \(\chi_{7623}(997,\cdot)\) \(\chi_{7623}(1174,\cdot)\) \(\chi_{7623}(1249,\cdot)\) \(\chi_{7623}(1300,\cdot)\) \(\chi_{7623}(1426,\cdot)\) \(\chi_{7623}(1438,\cdot)\) \(\chi_{7623}(1678,\cdot)\) \(\chi_{7623}(1690,\cdot)\) \(\chi_{7623}(1867,\cdot)\) \(\chi_{7623}(1942,\cdot)\) \(\chi_{7623}(1993,\cdot)\) \(\chi_{7623}(2119,\cdot)\) \(\chi_{7623}(2131,\cdot)\) \(\chi_{7623}(2257,\cdot)\) \(\chi_{7623}(2371,\cdot)\) \(\chi_{7623}(2383,\cdot)\) \(\chi_{7623}(2560,\cdot)\) \(\chi_{7623}(2686,\cdot)\) \(\chi_{7623}(2812,\cdot)\) \(\chi_{7623}(2824,\cdot)\) ...

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

\((848,4357,4600)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{6}\right),e\left(\frac{103}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1867, a) \) \(1\)\(1\)\(e\left(\frac{103}{110}\right)\)\(e\left(\frac{48}{55}\right)\)\(e\left(\frac{41}{330}\right)\)\(e\left(\frac{89}{110}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{122}{165}\right)\)\(e\left(\frac{41}{55}\right)\)\(e\left(\frac{118}{165}\right)\)\(e\left(\frac{146}{165}\right)\)\(e\left(\frac{329}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1867,a) \;\) at \(\;a = \) e.g. 2