Properties

Label 4029.970
Modulus $4029$
Conductor $79$
Order $13$
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(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,24]))
 
pari: [g,chi] = znchar(Mod(970,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(79\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(13\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{79}(22,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.bc

\(\chi_{4029}(52,\cdot)\) \(\chi_{4029}(460,\cdot)\) \(\chi_{4029}(970,\cdot)\) \(\chi_{4029}(2092,\cdot)\) \(\chi_{4029}(2143,\cdot)\) \(\chi_{4029}(2653,\cdot)\) \(\chi_{4029}(2704,\cdot)\) \(\chi_{4029}(2908,\cdot)\) \(\chi_{4029}(3010,\cdot)\) \(\chi_{4029}(3418,\cdot)\) \(\chi_{4029}(3622,\cdot)\) \(\chi_{4029}(3775,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 13 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(970, a) \) \(1\)\(1\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{10}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(970,a) \;\) at \(\;a = \) e.g. 2