Properties

Label 8034.4391
Modulus $8034$
Conductor $4017$
Order $102$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, base_ring=CyclotomicField(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([51,85,73]))
 
pari: [g,chi] = znchar(Mod(4391,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(4017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4017}(374,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.dm

\(\chi_{8034}(257,\cdot)\) \(\chi_{8034}(329,\cdot)\) \(\chi_{8034}(563,\cdot)\) \(\chi_{8034}(719,\cdot)\) \(\chi_{8034}(1187,\cdot)\) \(\chi_{8034}(1427,\cdot)\) \(\chi_{8034}(1733,\cdot)\) \(\chi_{8034}(1889,\cdot)\) \(\chi_{8034}(2825,\cdot)\) \(\chi_{8034}(3143,\cdot)\) \(\chi_{8034}(3371,\cdot)\) \(\chi_{8034}(3683,\cdot)\) \(\chi_{8034}(4229,\cdot)\) \(\chi_{8034}(4235,\cdot)\) \(\chi_{8034}(4307,\cdot)\) \(\chi_{8034}(4391,\cdot)\) \(\chi_{8034}(4469,\cdot)\) \(\chi_{8034}(4619,\cdot)\) \(\chi_{8034}(4697,\cdot)\) \(\chi_{8034}(4937,\cdot)\) \(\chi_{8034}(5015,\cdot)\) \(\chi_{8034}(5171,\cdot)\) \(\chi_{8034}(5249,\cdot)\) \(\chi_{8034}(5399,\cdot)\) \(\chi_{8034}(5945,\cdot)\) \(\chi_{8034}(6185,\cdot)\) \(\chi_{8034}(6257,\cdot)\) \(\chi_{8034}(6575,\cdot)\) \(\chi_{8034}(6809,\cdot)\) \(\chi_{8034}(7277,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((5357,1237,5773)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{73}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(4391, a) \) \(1\)\(1\)\(e\left(\frac{73}{102}\right)\)\(e\left(\frac{1}{34}\right)\)\(e\left(\frac{101}{102}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{43}{102}\right)\)\(e\left(\frac{1}{102}\right)\)\(e\left(\frac{22}{51}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{38}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(4391,a) \;\) at \(\;a = \) e.g. 2