Properties

Label 4029.701
Modulus $4029$
Conductor $4029$
Order $52$
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(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,39,18]))
 
pari: [g,chi] = znchar(Mod(701,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(4029\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
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.bx

\(\chi_{4029}(140,\cdot)\) \(\chi_{4029}(191,\cdot)\) \(\chi_{4029}(251,\cdot)\) \(\chi_{4029}(659,\cdot)\) \(\chi_{4029}(701,\cdot)\) \(\chi_{4029}(752,\cdot)\) \(\chi_{4029}(965,\cdot)\) \(\chi_{4029}(1118,\cdot)\) \(\chi_{4029}(1322,\cdot)\) \(\chi_{4029}(1730,\cdot)\) \(\chi_{4029}(1832,\cdot)\) \(\chi_{4029}(1874,\cdot)\) \(\chi_{4029}(2036,\cdot)\) \(\chi_{4029}(2087,\cdot)\) \(\chi_{4029}(2384,\cdot)\) \(\chi_{4029}(2597,\cdot)\) \(\chi_{4029}(2648,\cdot)\) \(\chi_{4029}(2792,\cdot)\) \(\chi_{4029}(3098,\cdot)\) \(\chi_{4029}(3251,\cdot)\) \(\chi_{4029}(3455,\cdot)\) \(\chi_{4029}(3770,\cdot)\) \(\chi_{4029}(3863,\cdot)\) \(\chi_{4029}(3965,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((2687,3556,3163)\) → \((-1,-i,e\left(\frac{9}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(701, a) \) \(1\)\(1\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{37}{52}\right)\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{5}{52}\right)\)\(e\left(\frac{15}{52}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{51}{52}\right)\)\(e\left(\frac{7}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(701,a) \;\) at \(\;a = \) e.g. 2