Properties

Label 4028.197
Modulus $4028$
Conductor $1007$
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(4028, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,26,57]))
 
pari: [g,chi] = znchar(Mod(197,4028))
 

Basic properties

Modulus: \(4028\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1007}(197,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4028.bx

\(\chi_{4028}(197,\cdot)\) \(\chi_{4028}(653,\cdot)\) \(\chi_{4028}(729,\cdot)\) \(\chi_{4028}(885,\cdot)\) \(\chi_{4028}(961,\cdot)\) \(\chi_{4028}(1493,\cdot)\) \(\chi_{4028}(1721,\cdot)\) \(\chi_{4028}(1945,\cdot)\) \(\chi_{4028}(2021,\cdot)\) \(\chi_{4028}(2025,\cdot)\) \(\chi_{4028}(2177,\cdot)\) \(\chi_{4028}(2481,\cdot)\) \(\chi_{4028}(2553,\cdot)\) \(\chi_{4028}(2709,\cdot)\) \(\chi_{4028}(2781,\cdot)\) \(\chi_{4028}(2785,\cdot)\) \(\chi_{4028}(3085,\cdot)\) \(\chi_{4028}(3165,\cdot)\) \(\chi_{4028}(3237,\cdot)\) \(\chi_{4028}(3541,\cdot)\) \(\chi_{4028}(3621,\cdot)\) \(\chi_{4028}(3697,\cdot)\) \(\chi_{4028}(3769,\cdot)\) \(\chi_{4028}(3845,\cdot)\)

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

\((2015,2757,2281)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{19}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 4028 }(197, a) \) \(1\)\(1\)\(e\left(\frac{59}{78}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4028 }(197,a) \;\) at \(\;a = \) e.g. 2