Properties

Label 4029.77
Modulus $4029$
Conductor $4029$
Order $312$
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(312))
 
M = H._module
 
chi = DirichletCharacter(H, M([156,39,172]))
 
pari: [g,chi] = znchar(Mod(77,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(4029\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(312\)
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.cw

\(\chi_{4029}(53,\cdot)\) \(\chi_{4029}(59,\cdot)\) \(\chi_{4029}(77,\cdot)\) \(\chi_{4029}(161,\cdot)\) \(\chi_{4029}(206,\cdot)\) \(\chi_{4029}(212,\cdot)\) \(\chi_{4029}(314,\cdot)\) \(\chi_{4029}(359,\cdot)\) \(\chi_{4029}(434,\cdot)\) \(\chi_{4029}(461,\cdot)\) \(\chi_{4029}(542,\cdot)\) \(\chi_{4029}(587,\cdot)\) \(\chi_{4029}(638,\cdot)\) \(\chi_{4029}(671,\cdot)\) \(\chi_{4029}(695,\cdot)\) \(\chi_{4029}(740,\cdot)\) \(\chi_{4029}(746,\cdot)\) \(\chi_{4029}(797,\cdot)\) \(\chi_{4029}(818,\cdot)\) \(\chi_{4029}(824,\cdot)\) \(\chi_{4029}(875,\cdot)\) \(\chi_{4029}(899,\cdot)\) \(\chi_{4029}(944,\cdot)\) \(\chi_{4029}(977,\cdot)\) \(\chi_{4029}(995,\cdot)\) \(\chi_{4029}(1001,\cdot)\) \(\chi_{4029}(1022,\cdot)\) \(\chi_{4029}(1097,\cdot)\) \(\chi_{4029}(1154,\cdot)\) \(\chi_{4029}(1181,\cdot)\) ...

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: $\Q(\zeta_{312})$
Fixed field: Number field defined by a degree 312 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{43}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(77, a) \) \(1\)\(1\)\(e\left(\frac{71}{156}\right)\)\(e\left(\frac{71}{78}\right)\)\(e\left(\frac{95}{312}\right)\)\(e\left(\frac{185}{312}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{79}{104}\right)\)\(e\left(\frac{269}{312}\right)\)\(e\left(\frac{19}{78}\right)\)\(e\left(\frac{5}{104}\right)\)\(e\left(\frac{32}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(77,a) \;\) at \(\;a = \) e.g. 2