Properties

Label 3997.17
Modulus $3997$
Conductor $3997$
Order $570$
Real no
Primitive yes
Minimal yes
Parity even

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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([95,413]))
 
pari: [g,chi] = znchar(Mod(17,3997))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3997.ct

\(\chi_{3997}(17,\cdot)\) \(\chi_{3997}(19,\cdot)\) \(\chi_{3997}(101,\cdot)\) \(\chi_{3997}(234,\cdot)\) \(\chi_{3997}(276,\cdot)\) \(\chi_{3997}(278,\cdot)\) \(\chi_{3997}(348,\cdot)\) \(\chi_{3997}(369,\cdot)\) \(\chi_{3997}(381,\cdot)\) \(\chi_{3997}(402,\cdot)\) \(\chi_{3997}(416,\cdot)\) \(\chi_{3997}(430,\cdot)\) \(\chi_{3997}(474,\cdot)\) \(\chi_{3997}(488,\cdot)\) \(\chi_{3997}(514,\cdot)\) \(\chi_{3997}(537,\cdot)\) \(\chi_{3997}(558,\cdot)\) \(\chi_{3997}(647,\cdot)\) \(\chi_{3997}(684,\cdot)\) \(\chi_{3997}(822,\cdot)\) \(\chi_{3997}(878,\cdot)\) \(\chi_{3997}(915,\cdot)\) \(\chi_{3997}(920,\cdot)\) \(\chi_{3997}(957,\cdot)\) \(\chi_{3997}(962,\cdot)\) \(\chi_{3997}(964,\cdot)\) \(\chi_{3997}(983,\cdot)\) \(\chi_{3997}(985,\cdot)\) \(\chi_{3997}(1046,\cdot)\) \(\chi_{3997}(1062,\cdot)\) ...

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

\((2285,1716)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{413}{570}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 3997 }(17, a) \) \(1\)\(1\)\(e\left(\frac{25}{114}\right)\)\(e\left(\frac{254}{285}\right)\)\(e\left(\frac{25}{57}\right)\)\(e\left(\frac{341}{570}\right)\)\(e\left(\frac{21}{190}\right)\)\(e\left(\frac{25}{38}\right)\)\(e\left(\frac{223}{285}\right)\)\(e\left(\frac{233}{285}\right)\)\(e\left(\frac{242}{285}\right)\)\(e\left(\frac{94}{285}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3997 }(17,a) \;\) at \(\;a = \) e.g. 2