Properties

Label 4029.3943
Modulus $4029$
Conductor $1343$
Order $78$
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([0,39,14]))
 
pari: [g,chi] = znchar(Mod(3943,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(1343\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1343}(1257,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.cf

\(\chi_{4029}(16,\cdot)\) \(\chi_{4029}(169,\cdot)\) \(\chi_{4029}(730,\cdot)\) \(\chi_{4029}(832,\cdot)\) \(\chi_{4029}(1036,\cdot)\) \(\chi_{4029}(1138,\cdot)\) \(\chi_{4029}(1189,\cdot)\) \(\chi_{4029}(1393,\cdot)\) \(\chi_{4029}(1495,\cdot)\) \(\chi_{4029}(1546,\cdot)\) \(\chi_{4029}(1699,\cdot)\) \(\chi_{4029}(2056,\cdot)\) \(\chi_{4029}(2158,\cdot)\) \(\chi_{4029}(2209,\cdot)\) \(\chi_{4029}(2311,\cdot)\) \(\chi_{4029}(2770,\cdot)\) \(\chi_{4029}(2974,\cdot)\) \(\chi_{4029}(3331,\cdot)\) \(\chi_{4029}(3433,\cdot)\) \(\chi_{4029}(3586,\cdot)\) \(\chi_{4029}(3739,\cdot)\) \(\chi_{4029}(3841,\cdot)\) \(\chi_{4029}(3943,\cdot)\) \(\chi_{4029}(3994,\cdot)\)

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 78 polynomial

Values on generators

\((2687,3556,3163)\) → \((1,-1,e\left(\frac{7}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(3943, a) \) \(1\)\(1\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{34}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(3943,a) \;\) at \(\;a = \) e.g. 2