Properties

Label 4029.259
Modulus $4029$
Conductor $1343$
Order $52$
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(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,39,48]))
 
pari: [g,chi] = znchar(Mod(259,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(1343\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1343}(259,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.bz

\(\chi_{4029}(64,\cdot)\) \(\chi_{4029}(166,\cdot)\) \(\chi_{4029}(259,\cdot)\) \(\chi_{4029}(574,\cdot)\) \(\chi_{4029}(778,\cdot)\) \(\chi_{4029}(931,\cdot)\) \(\chi_{4029}(1237,\cdot)\) \(\chi_{4029}(1381,\cdot)\) \(\chi_{4029}(1432,\cdot)\) \(\chi_{4029}(1645,\cdot)\) \(\chi_{4029}(1942,\cdot)\) \(\chi_{4029}(1993,\cdot)\) \(\chi_{4029}(2155,\cdot)\) \(\chi_{4029}(2197,\cdot)\) \(\chi_{4029}(2299,\cdot)\) \(\chi_{4029}(2707,\cdot)\) \(\chi_{4029}(2911,\cdot)\) \(\chi_{4029}(3064,\cdot)\) \(\chi_{4029}(3277,\cdot)\) \(\chi_{4029}(3328,\cdot)\) \(\chi_{4029}(3370,\cdot)\) \(\chi_{4029}(3778,\cdot)\) \(\chi_{4029}(3838,\cdot)\) \(\chi_{4029}(3889,\cdot)\)

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

\((2687,3556,3163)\) → \((1,-i,e\left(\frac{12}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(259, a) \) \(1\)\(1\)\(e\left(\frac{5}{26}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{51}{52}\right)\)\(e\left(\frac{9}{52}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{9}{52}\right)\)\(e\left(\frac{1}{52}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{10}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(259,a) \;\) at \(\;a = \) e.g. 2