Properties

Label 4017.32
Modulus $4017$
Conductor $4017$
Order $204$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4017, base_ring=CyclotomicField(204))
 
M = H._module
 
chi = DirichletCharacter(H, M([102,85,32]))
 
pari: [g,chi] = znchar(Mod(32,4017))
 

Basic properties

Modulus: \(4017\)
Conductor: \(4017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(204\)
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 4017.ee

\(\chi_{4017}(2,\cdot)\) \(\chi_{4017}(32,\cdot)\) \(\chi_{4017}(41,\cdot)\) \(\chi_{4017}(98,\cdot)\) \(\chi_{4017}(110,\cdot)\) \(\chi_{4017}(119,\cdot)\) \(\chi_{4017}(128,\cdot)\) \(\chi_{4017}(158,\cdot)\) \(\chi_{4017}(392,\cdot)\) \(\chi_{4017}(431,\cdot)\) \(\chi_{4017}(440,\cdot)\) \(\chi_{4017}(509,\cdot)\) \(\chi_{4017}(578,\cdot)\) \(\chi_{4017}(656,\cdot)\) \(\chi_{4017}(773,\cdot)\) \(\chi_{4017}(812,\cdot)\) \(\chi_{4017}(839,\cdot)\) \(\chi_{4017}(986,\cdot)\) \(\chi_{4017}(1034,\cdot)\) \(\chi_{4017}(1055,\cdot)\) \(\chi_{4017}(1112,\cdot)\) \(\chi_{4017}(1268,\cdot)\) \(\chi_{4017}(1346,\cdot)\) \(\chi_{4017}(1367,\cdot)\) \(\chi_{4017}(1397,\cdot)\) \(\chi_{4017}(1475,\cdot)\) \(\chi_{4017}(1571,\cdot)\) \(\chi_{4017}(1787,\cdot)\) \(\chi_{4017}(1814,\cdot)\) \(\chi_{4017}(1883,\cdot)\) ...

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: $\Q(\zeta_{204})$
Fixed field: Number field defined by a degree 204 polynomial (not computed)

Values on generators

\((1340,1237,1756)\) → \((-1,e\left(\frac{5}{12}\right),e\left(\frac{8}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 4017 }(32, a) \) \(1\)\(1\)\(e\left(\frac{167}{204}\right)\)\(e\left(\frac{65}{102}\right)\)\(e\left(\frac{83}{204}\right)\)\(e\left(\frac{43}{204}\right)\)\(e\left(\frac{31}{68}\right)\)\(e\left(\frac{23}{102}\right)\)\(e\left(\frac{67}{68}\right)\)\(e\left(\frac{1}{34}\right)\)\(e\left(\frac{14}{51}\right)\)\(e\left(\frac{16}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4017 }(32,a) \;\) at \(\;a = \) e.g. 2