Properties

Label 3997.13
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([285,352]))
 
pari: [g,chi] = znchar(Mod(13,3997))
 

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.cz

\(\chi_{3997}(13,\cdot)\) \(\chi_{3997}(34,\cdot)\) \(\chi_{3997}(83,\cdot)\) \(\chi_{3997}(97,\cdot)\) \(\chi_{3997}(139,\cdot)\) \(\chi_{3997}(174,\cdot)\) \(\chi_{3997}(202,\cdot)\) \(\chi_{3997}(223,\cdot)\) \(\chi_{3997}(237,\cdot)\) \(\chi_{3997}(244,\cdot)\) \(\chi_{3997}(279,\cdot)\) \(\chi_{3997}(293,\cdot)\) \(\chi_{3997}(321,\cdot)\) \(\chi_{3997}(342,\cdot)\) \(\chi_{3997}(482,\cdot)\) \(\chi_{3997}(531,\cdot)\) \(\chi_{3997}(538,\cdot)\) \(\chi_{3997}(552,\cdot)\) \(\chi_{3997}(559,\cdot)\) \(\chi_{3997}(580,\cdot)\) \(\chi_{3997}(601,\cdot)\) \(\chi_{3997}(608,\cdot)\) \(\chi_{3997}(615,\cdot)\) \(\chi_{3997}(671,\cdot)\) \(\chi_{3997}(692,\cdot)\) \(\chi_{3997}(762,\cdot)\) \(\chi_{3997}(790,\cdot)\) \(\chi_{3997}(804,\cdot)\) \(\chi_{3997}(832,\cdot)\) \(\chi_{3997}(846,\cdot)\) ...

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

\((2285,1716)\) → \((-1,e\left(\frac{176}{285}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 3997 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{32}{57}\right)\)\(e\left(\frac{67}{570}\right)\)\(e\left(\frac{7}{57}\right)\)\(e\left(\frac{59}{570}\right)\)\(e\left(\frac{129}{190}\right)\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{67}{285}\right)\)\(e\left(\frac{379}{570}\right)\)\(e\left(\frac{143}{285}\right)\)\(e\left(\frac{137}{570}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3997 }(13,a) \;\) at \(\;a = \) e.g. 2