Properties

Label 4029.109
Modulus $4029$
Conductor $1343$
Order $624$
Real no
Primitive no
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([0,429,536]))
 
pari: [g,chi] = znchar(Mod(109,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(1343\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(624\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1343}(109,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.cy

\(\chi_{4029}(7,\cdot)\) \(\chi_{4029}(28,\cdot)\) \(\chi_{4029}(37,\cdot)\) \(\chi_{4029}(82,\cdot)\) \(\chi_{4029}(109,\cdot)\) \(\chi_{4029}(133,\cdot)\) \(\chi_{4029}(139,\cdot)\) \(\chi_{4029}(142,\cdot)\) \(\chi_{4029}(193,\cdot)\) \(\chi_{4029}(211,\cdot)\) \(\chi_{4029}(226,\cdot)\) \(\chi_{4029}(232,\cdot)\) \(\chi_{4029}(235,\cdot)\) \(\chi_{4029}(244,\cdot)\) \(\chi_{4029}(265,\cdot)\) \(\chi_{4029}(346,\cdot)\) \(\chi_{4029}(364,\cdot)\) \(\chi_{4029}(379,\cdot)\) \(\chi_{4029}(430,\cdot)\) \(\chi_{4029}(448,\cdot)\) \(\chi_{4029}(454,\cdot)\) \(\chi_{4029}(469,\cdot)\) \(\chi_{4029}(481,\cdot)\) \(\chi_{4029}(517,\cdot)\) \(\chi_{4029}(556,\cdot)\) \(\chi_{4029}(583,\cdot)\) \(\chi_{4029}(592,\cdot)\) \(\chi_{4029}(601,\cdot)\) \(\chi_{4029}(607,\cdot)\) \(\chi_{4029}(619,\cdot)\) ...

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

\((2687,3556,3163)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{67}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(109, a) \) \(1\)\(1\)\(e\left(\frac{19}{312}\right)\)\(e\left(\frac{19}{156}\right)\)\(e\left(\frac{433}{624}\right)\)\(e\left(\frac{55}{624}\right)\)\(e\left(\frac{19}{104}\right)\)\(e\left(\frac{157}{208}\right)\)\(e\left(\frac{139}{624}\right)\)\(e\left(\frac{149}{156}\right)\)\(e\left(\frac{31}{208}\right)\)\(e\left(\frac{19}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(109,a) \;\) at \(\;a = \) e.g. 2