Properties

Label 4023.19
Modulus $4023$
Conductor $1341$
Order $111$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4023, base_ring=CyclotomicField(222))
 
M = H._module
 
chi = DirichletCharacter(H, M([148,126]))
 
pari: [g,chi] = znchar(Mod(19,4023))
 

Basic properties

Modulus: \(4023\)
Conductor: \(1341\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(111\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1341}(466,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4023.w

\(\chi_{4023}(19,\cdot)\) \(\chi_{4023}(37,\cdot)\) \(\chi_{4023}(46,\cdot)\) \(\chi_{4023}(73,\cdot)\) \(\chi_{4023}(127,\cdot)\) \(\chi_{4023}(145,\cdot)\) \(\chi_{4023}(154,\cdot)\) \(\chi_{4023}(253,\cdot)\) \(\chi_{4023}(289,\cdot)\) \(\chi_{4023}(334,\cdot)\) \(\chi_{4023}(361,\cdot)\) \(\chi_{4023}(478,\cdot)\) \(\chi_{4023}(496,\cdot)\) \(\chi_{4023}(532,\cdot)\) \(\chi_{4023}(613,\cdot)\) \(\chi_{4023}(721,\cdot)\) \(\chi_{4023}(775,\cdot)\) \(\chi_{4023}(847,\cdot)\) \(\chi_{4023}(874,\cdot)\) \(\chi_{4023}(910,\cdot)\) \(\chi_{4023}(982,\cdot)\) \(\chi_{4023}(1036,\cdot)\) \(\chi_{4023}(1072,\cdot)\) \(\chi_{4023}(1198,\cdot)\) \(\chi_{4023}(1225,\cdot)\) \(\chi_{4023}(1288,\cdot)\) \(\chi_{4023}(1306,\cdot)\) \(\chi_{4023}(1315,\cdot)\) \(\chi_{4023}(1360,\cdot)\) \(\chi_{4023}(1369,\cdot)\) ...

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: $\Q(\zeta_{111})$
Fixed field: Number field defined by a degree 111 polynomial (not computed)

Values on generators

\((299,3727)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{21}{37}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4023 }(19, a) \) \(1\)\(1\)\(e\left(\frac{26}{111}\right)\)\(e\left(\frac{52}{111}\right)\)\(e\left(\frac{40}{111}\right)\)\(e\left(\frac{29}{111}\right)\)\(e\left(\frac{26}{37}\right)\)\(e\left(\frac{22}{37}\right)\)\(e\left(\frac{59}{111}\right)\)\(e\left(\frac{46}{111}\right)\)\(e\left(\frac{55}{111}\right)\)\(e\left(\frac{104}{111}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4023 }(19,a) \;\) at \(\;a = \) e.g. 2