Properties

Label 4023.20
Modulus $4023$
Conductor $4023$
Order $666$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4023, base_ring=CyclotomicField(666))
 
M = H._module
 
chi = DirichletCharacter(H, M([259,477]))
 
pari: [g,chi] = znchar(Mod(20,4023))
 

Basic properties

Modulus: \(4023\)
Conductor: \(4023\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(666\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4023.bh

\(\chi_{4023}(20,\cdot)\) \(\chi_{4023}(47,\cdot)\) \(\chi_{4023}(68,\cdot)\) \(\chi_{4023}(86,\cdot)\) \(\chi_{4023}(110,\cdot)\) \(\chi_{4023}(113,\cdot)\) \(\chi_{4023}(119,\cdot)\) \(\chi_{4023}(158,\cdot)\) \(\chi_{4023}(173,\cdot)\) \(\chi_{4023}(191,\cdot)\) \(\chi_{4023}(194,\cdot)\) \(\chi_{4023}(203,\cdot)\) \(\chi_{4023}(218,\cdot)\) \(\chi_{4023}(281,\cdot)\) \(\chi_{4023}(293,\cdot)\) \(\chi_{4023}(302,\cdot)\) \(\chi_{4023}(320,\cdot)\) \(\chi_{4023}(362,\cdot)\) \(\chi_{4023}(374,\cdot)\) \(\chi_{4023}(380,\cdot)\) \(\chi_{4023}(398,\cdot)\) \(\chi_{4023}(401,\cdot)\) \(\chi_{4023}(410,\cdot)\) \(\chi_{4023}(416,\cdot)\) \(\chi_{4023}(419,\cdot)\) \(\chi_{4023}(428,\cdot)\) \(\chi_{4023}(473,\cdot)\) \(\chi_{4023}(482,\cdot)\) \(\chi_{4023}(500,\cdot)\) \(\chi_{4023}(515,\cdot)\) ...

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: $\Q(\zeta_{333})$
Fixed field: Number field defined by a degree 666 polynomial (not computed)

Values on generators

\((299,3727)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{53}{74}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4023 }(20, a) \) \(-1\)\(1\)\(e\left(\frac{35}{333}\right)\)\(e\left(\frac{70}{333}\right)\)\(e\left(\frac{287}{666}\right)\)\(e\left(\frac{308}{333}\right)\)\(e\left(\frac{35}{111}\right)\)\(e\left(\frac{119}{222}\right)\)\(e\left(\frac{41}{333}\right)\)\(e\left(\frac{47}{666}\right)\)\(e\left(\frac{10}{333}\right)\)\(e\left(\frac{140}{333}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4023 }(20,a) \;\) at \(\;a = \) e.g. 2