Properties

Label 4029.43
Modulus $4029$
Conductor $1343$
Order $312$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(312))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,39,196]))
 
pari: [g,chi] = znchar(Mod(43,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(1343\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(312\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1343}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.cv

\(\chi_{4029}(43,\cdot)\) \(\chi_{4029}(70,\cdot)\) \(\chi_{4029}(127,\cdot)\) \(\chi_{4029}(145,\cdot)\) \(\chi_{4029}(274,\cdot)\) \(\chi_{4029}(280,\cdot)\) \(\chi_{4029}(355,\cdot)\) \(\chi_{4029}(376,\cdot)\) \(\chi_{4029}(382,\cdot)\) \(\chi_{4029}(502,\cdot)\) \(\chi_{4029}(508,\cdot)\) \(\chi_{4029}(559,\cdot)\) \(\chi_{4029}(661,\cdot)\) \(\chi_{4029}(706,\cdot)\) \(\chi_{4029}(739,\cdot)\) \(\chi_{4029}(865,\cdot)\) \(\chi_{4029}(916,\cdot)\) \(\chi_{4029}(937,\cdot)\) \(\chi_{4029}(943,\cdot)\) \(\chi_{4029}(1018,\cdot)\) \(\chi_{4029}(1090,\cdot)\) \(\chi_{4029}(1141,\cdot)\) \(\chi_{4029}(1165,\cdot)\) \(\chi_{4029}(1192,\cdot)\) \(\chi_{4029}(1222,\cdot)\) \(\chi_{4029}(1267,\cdot)\) \(\chi_{4029}(1294,\cdot)\) \(\chi_{4029}(1318,\cdot)\) \(\chi_{4029}(1324,\cdot)\) \(\chi_{4029}(1396,\cdot)\) ...

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: $\Q(\zeta_{312})$
Fixed field: Number field defined by a degree 312 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{49}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{41}{156}\right)\)\(e\left(\frac{41}{78}\right)\)\(e\left(\frac{179}{312}\right)\)\(e\left(\frac{209}{312}\right)\)\(e\left(\frac{41}{52}\right)\)\(e\left(\frac{87}{104}\right)\)\(e\left(\frac{185}{312}\right)\)\(e\left(\frac{67}{78}\right)\)\(e\left(\frac{97}{104}\right)\)\(e\left(\frac{2}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(43,a) \;\) at \(\;a = \) e.g. 2