Properties

Label 4029.1475
Modulus $4029$
Conductor $4029$
Order $156$
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(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,39,154]))
 
pari: [g,chi] = znchar(Mod(1475,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(4029\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
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.cp

\(\chi_{4029}(47,\cdot)\) \(\chi_{4029}(149,\cdot)\) \(\chi_{4029}(344,\cdot)\) \(\chi_{4029}(353,\cdot)\) \(\chi_{4029}(455,\cdot)\) \(\chi_{4029}(548,\cdot)\) \(\chi_{4029}(1007,\cdot)\) \(\chi_{4029}(1016,\cdot)\) \(\chi_{4029}(1109,\cdot)\) \(\chi_{4029}(1160,\cdot)\) \(\chi_{4029}(1169,\cdot)\) \(\chi_{4029}(1220,\cdot)\) \(\chi_{4029}(1262,\cdot)\) \(\chi_{4029}(1271,\cdot)\) \(\chi_{4029}(1373,\cdot)\) \(\chi_{4029}(1475,\cdot)\) \(\chi_{4029}(1619,\cdot)\) \(\chi_{4029}(1628,\cdot)\) \(\chi_{4029}(1772,\cdot)\) \(\chi_{4029}(1781,\cdot)\) \(\chi_{4029}(1823,\cdot)\) \(\chi_{4029}(1883,\cdot)\) \(\chi_{4029}(1925,\cdot)\) \(\chi_{4029}(2129,\cdot)\) \(\chi_{4029}(2180,\cdot)\) \(\chi_{4029}(2240,\cdot)\) \(\chi_{4029}(2282,\cdot)\) \(\chi_{4029}(2444,\cdot)\) \(\chi_{4029}(2486,\cdot)\) \(\chi_{4029}(2588,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((-1,i,e\left(\frac{77}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(1475, a) \) \(1\)\(1\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{149}{156}\right)\)\(e\left(\frac{11}{156}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{47}{52}\right)\)\(e\left(\frac{59}{156}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{1}{52}\right)\)\(e\left(\frac{31}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(1475,a) \;\) at \(\;a = \) e.g. 2