Properties

Label 8034.11
Modulus $8034$
Conductor $4017$
Order $204$
Real no
Primitive no
Minimal yes
Parity odd

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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([102,119,122]))
 
pari: [g,chi] = znchar(Mod(11,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(4017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(204\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4017}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.eb

\(\chi_{8034}(11,\cdot)\) \(\chi_{8034}(293,\cdot)\) \(\chi_{8034}(371,\cdot)\) \(\chi_{8034}(383,\cdot)\) \(\chi_{8034}(479,\cdot)\) \(\chi_{8034}(695,\cdot)\) \(\chi_{8034}(791,\cdot)\) \(\chi_{8034}(869,\cdot)\) \(\chi_{8034}(1073,\cdot)\) \(\chi_{8034}(1289,\cdot)\) \(\chi_{8034}(1493,\cdot)\) \(\chi_{8034}(1619,\cdot)\) \(\chi_{8034}(1931,\cdot)\) \(\chi_{8034}(2381,\cdot)\) \(\chi_{8034}(2537,\cdot)\) \(\chi_{8034}(2615,\cdot)\) \(\chi_{8034}(2663,\cdot)\) \(\chi_{8034}(2801,\cdot)\) \(\chi_{8034}(3035,\cdot)\) \(\chi_{8034}(3083,\cdot)\) \(\chi_{8034}(3161,\cdot)\) \(\chi_{8034}(3191,\cdot)\) \(\chi_{8034}(3317,\cdot)\) \(\chi_{8034}(3395,\cdot)\) \(\chi_{8034}(3659,\cdot)\) \(\chi_{8034}(4037,\cdot)\) \(\chi_{8034}(4205,\cdot)\) \(\chi_{8034}(4271,\cdot)\) \(\chi_{8034}(4331,\cdot)\) \(\chi_{8034}(4361,\cdot)\) ...

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

\((5357,1237,5773)\) → \((-1,e\left(\frac{7}{12}\right),e\left(\frac{61}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{71}{204}\right)\)\(e\left(\frac{55}{68}\right)\)\(e\left(\frac{13}{204}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{155}{204}\right)\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{71}{102}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{23}{68}\right)\)\(e\left(\frac{8}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(11,a) \;\) at \(\;a = \) e.g. 2