Properties

Label 3997.38
Modulus $3997$
Conductor $3997$
Order $570$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3997, base_ring=CyclotomicField(570))
 
M = H._module
 
chi = DirichletCharacter(H, M([95,432]))
 
pari: [g,chi] = znchar(Mod(38,3997))
 

Basic properties

Modulus: \(3997\)
Conductor: \(3997\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(570\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3997.cx

\(\chi_{3997}(38,\cdot)\) \(\chi_{3997}(117,\cdot)\) \(\chi_{3997}(138,\cdot)\) \(\chi_{3997}(201,\cdot)\) \(\chi_{3997}(206,\cdot)\) \(\chi_{3997}(208,\cdot)\) \(\chi_{3997}(215,\cdot)\) \(\chi_{3997}(257,\cdot)\) \(\chi_{3997}(283,\cdot)\) \(\chi_{3997}(409,\cdot)\) \(\chi_{3997}(411,\cdot)\) \(\chi_{3997}(439,\cdot)\) \(\chi_{3997}(460,\cdot)\) \(\chi_{3997}(521,\cdot)\) \(\chi_{3997}(523,\cdot)\) \(\chi_{3997}(544,\cdot)\) \(\chi_{3997}(549,\cdot)\) \(\chi_{3997}(556,\cdot)\) \(\chi_{3997}(577,\cdot)\) \(\chi_{3997}(591,\cdot)\) \(\chi_{3997}(607,\cdot)\) \(\chi_{3997}(691,\cdot)\) \(\chi_{3997}(696,\cdot)\) \(\chi_{3997}(717,\cdot)\) \(\chi_{3997}(719,\cdot)\) \(\chi_{3997}(747,\cdot)\) \(\chi_{3997}(787,\cdot)\) \(\chi_{3997}(796,\cdot)\) \(\chi_{3997}(838,\cdot)\) \(\chi_{3997}(852,\cdot)\) ...

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: $\Q(\zeta_{285})$
Fixed field: Number field defined by a degree 570 polynomial (not computed)

Values on generators

\((2285,1716)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{72}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 3997 }(38, a) \) \(-1\)\(1\)\(e\left(\frac{22}{57}\right)\)\(e\left(\frac{527}{570}\right)\)\(e\left(\frac{44}{57}\right)\)\(e\left(\frac{379}{570}\right)\)\(e\left(\frac{59}{190}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{242}{285}\right)\)\(e\left(\frac{29}{570}\right)\)\(e\left(\frac{223}{285}\right)\)\(e\left(\frac{397}{570}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3997 }(38,a) \;\) at \(\;a = \) e.g. 2