Properties

Label 8034.733
Modulus $8034$
Conductor $1339$
Order $204$
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(204))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,153,50]))
 
pari: [g,chi] = znchar(Mod(733,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(204\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1339}(733,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.eg

\(\chi_{8034}(109,\cdot)\) \(\chi_{8034}(151,\cdot)\) \(\chi_{8034}(187,\cdot)\) \(\chi_{8034}(307,\cdot)\) \(\chi_{8034}(463,\cdot)\) \(\chi_{8034}(499,\cdot)\) \(\chi_{8034}(577,\cdot)\) \(\chi_{8034}(733,\cdot)\) \(\chi_{8034}(775,\cdot)\) \(\chi_{8034}(889,\cdot)\) \(\chi_{8034}(967,\cdot)\) \(\chi_{8034}(1279,\cdot)\) \(\chi_{8034}(1321,\cdot)\) \(\chi_{8034}(1435,\cdot)\) \(\chi_{8034}(1477,\cdot)\) \(\chi_{8034}(1513,\cdot)\) \(\chi_{8034}(1633,\cdot)\) \(\chi_{8034}(1669,\cdot)\) \(\chi_{8034}(1747,\cdot)\) \(\chi_{8034}(1825,\cdot)\) \(\chi_{8034}(2137,\cdot)\) \(\chi_{8034}(2413,\cdot)\) \(\chi_{8034}(2683,\cdot)\) \(\chi_{8034}(2959,\cdot)\) \(\chi_{8034}(3073,\cdot)\) \(\chi_{8034}(3271,\cdot)\) \(\chi_{8034}(3307,\cdot)\) \(\chi_{8034}(3349,\cdot)\) \(\chi_{8034}(3775,\cdot)\) \(\chi_{8034}(3817,\cdot)\) ...

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

\((5357,1237,5773)\) → \((1,-i,e\left(\frac{25}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(733, a) \) \(1\)\(1\)\(e\left(\frac{203}{204}\right)\)\(e\left(\frac{47}{204}\right)\)\(e\left(\frac{41}{204}\right)\)\(e\left(\frac{67}{102}\right)\)\(e\left(\frac{73}{204}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{101}{102}\right)\)\(e\left(\frac{4}{51}\right)\)\(e\left(\frac{49}{68}\right)\)\(e\left(\frac{23}{102}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(733,a) \;\) at \(\;a = \) e.g. 2