Properties

Label 4029.715
Modulus $4029$
Conductor $79$
Order $39$
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(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,8]))
 
pari: [g,chi] = znchar(Mod(715,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(79\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{79}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.bs

\(\chi_{4029}(256,\cdot)\) \(\chi_{4029}(358,\cdot)\) \(\chi_{4029}(562,\cdot)\) \(\chi_{4029}(664,\cdot)\) \(\chi_{4029}(715,\cdot)\) \(\chi_{4029}(919,\cdot)\) \(\chi_{4029}(1021,\cdot)\) \(\chi_{4029}(1072,\cdot)\) \(\chi_{4029}(1225,\cdot)\) \(\chi_{4029}(1582,\cdot)\) \(\chi_{4029}(1684,\cdot)\) \(\chi_{4029}(1735,\cdot)\) \(\chi_{4029}(1837,\cdot)\) \(\chi_{4029}(2296,\cdot)\) \(\chi_{4029}(2500,\cdot)\) \(\chi_{4029}(2857,\cdot)\) \(\chi_{4029}(2959,\cdot)\) \(\chi_{4029}(3112,\cdot)\) \(\chi_{4029}(3265,\cdot)\) \(\chi_{4029}(3367,\cdot)\) \(\chi_{4029}(3469,\cdot)\) \(\chi_{4029}(3520,\cdot)\) \(\chi_{4029}(3571,\cdot)\) \(\chi_{4029}(3724,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\((2687,3556,3163)\) → \((1,1,e\left(\frac{4}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(715, a) \) \(1\)\(1\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{25}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(715,a) \;\) at \(\;a = \) e.g. 2