Properties

Label 4029.683
Modulus $4029$
Conductor $4029$
Order $624$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(624))
 
M = H._module
 
chi = DirichletCharacter(H, M([312,39,176]))
 
pari: [g,chi] = znchar(Mod(683,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(4029\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(624\)
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 4029.cz

\(\chi_{4029}(5,\cdot)\) \(\chi_{4029}(11,\cdot)\) \(\chi_{4029}(20,\cdot)\) \(\chi_{4029}(44,\cdot)\) \(\chi_{4029}(92,\cdot)\) \(\chi_{4029}(95,\cdot)\) \(\chi_{4029}(167,\cdot)\) \(\chi_{4029}(194,\cdot)\) \(\chi_{4029}(209,\cdot)\) \(\chi_{4029}(248,\cdot)\) \(\chi_{4029}(269,\cdot)\) \(\chi_{4029}(320,\cdot)\) \(\chi_{4029}(329,\cdot)\) \(\chi_{4029}(335,\cdot)\) \(\chi_{4029}(347,\cdot)\) \(\chi_{4029}(431,\cdot)\) \(\chi_{4029}(437,\cdot)\) \(\chi_{4029}(479,\cdot)\) \(\chi_{4029}(500,\cdot)\) \(\chi_{4029}(524,\cdot)\) \(\chi_{4029}(566,\cdot)\) \(\chi_{4029}(572,\cdot)\) \(\chi_{4029}(584,\cdot)\) \(\chi_{4029}(602,\cdot)\) \(\chi_{4029}(626,\cdot)\) \(\chi_{4029}(641,\cdot)\) \(\chi_{4029}(668,\cdot)\) \(\chi_{4029}(674,\cdot)\) \(\chi_{4029}(677,\cdot)\) \(\chi_{4029}(683,\cdot)\) ...

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: $\Q(\zeta_{624})$
Fixed field: Number field defined by a degree 624 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{11}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(683, a) \) \(1\)\(1\)\(e\left(\frac{157}{312}\right)\)\(e\left(\frac{1}{156}\right)\)\(e\left(\frac{187}{624}\right)\)\(e\left(\frac{397}{624}\right)\)\(e\left(\frac{53}{104}\right)\)\(e\left(\frac{167}{208}\right)\)\(e\left(\frac{73}{624}\right)\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{29}{208}\right)\)\(e\left(\frac{1}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(683,a) \;\) at \(\;a = \) e.g. 2