Properties

Label 8034.2789
Modulus $8034$
Conductor $4017$
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([102,187,60]))
 
pari: [g,chi] = znchar(Mod(2789,8034))
 

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 \(\chi_{4017}(2789,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.em

\(\chi_{8034}(137,\cdot)\) \(\chi_{8034}(167,\cdot)\) \(\chi_{8034}(215,\cdot)\) \(\chi_{8034}(323,\cdot)\) \(\chi_{8034}(587,\cdot)\) \(\chi_{8034}(821,\cdot)\) \(\chi_{8034}(917,\cdot)\) \(\chi_{8034}(1163,\cdot)\) \(\chi_{8034}(1259,\cdot)\) \(\chi_{8034}(1415,\cdot)\) \(\chi_{8034}(1523,\cdot)\) \(\chi_{8034}(1553,\cdot)\) \(\chi_{8034}(1709,\cdot)\) \(\chi_{8034}(1727,\cdot)\) \(\chi_{8034}(1991,\cdot)\) \(\chi_{8034}(2021,\cdot)\) \(\chi_{8034}(2069,\cdot)\) \(\chi_{8034}(2177,\cdot)\) \(\chi_{8034}(2399,\cdot)\) \(\chi_{8034}(2771,\cdot)\) \(\chi_{8034}(2789,\cdot)\) \(\chi_{8034}(2897,\cdot)\) \(\chi_{8034}(2945,\cdot)\) \(\chi_{8034}(3053,\cdot)\) \(\chi_{8034}(3113,\cdot)\) \(\chi_{8034}(3257,\cdot)\) \(\chi_{8034}(3269,\cdot)\) \(\chi_{8034}(3413,\cdot)\) \(\chi_{8034}(3581,\cdot)\) \(\chi_{8034}(3677,\cdot)\) ...

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

\((5357,1237,5773)\) → \((-1,e\left(\frac{11}{12}\right),e\left(\frac{5}{17}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(2789, a) \) \(1\)\(1\)\(e\left(\frac{3}{68}\right)\)\(e\left(\frac{53}{204}\right)\)\(e\left(\frac{175}{204}\right)\)\(e\left(\frac{47}{51}\right)\)\(e\left(\frac{23}{204}\right)\)\(e\left(\frac{37}{51}\right)\)\(e\left(\frac{3}{34}\right)\)\(e\left(\frac{47}{102}\right)\)\(e\left(\frac{1}{68}\right)\)\(e\left(\frac{31}{102}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(2789,a) \;\) at \(\;a = \) e.g. 2