Properties

Label 6004.35
Modulus $6004$
Conductor $6004$
Order $234$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6004, base_ring=CyclotomicField(234))
 
M = H._module
 
chi = DirichletCharacter(H, M([117,52,111]))
 
pari: [g,chi] = znchar(Mod(35,6004))
 

Basic properties

Modulus: \(6004\)
Conductor: \(6004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(234\)
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 6004.et

\(\chi_{6004}(35,\cdot)\) \(\chi_{6004}(43,\cdot)\) \(\chi_{6004}(47,\cdot)\) \(\chi_{6004}(139,\cdot)\) \(\chi_{6004}(195,\cdot)\) \(\chi_{6004}(271,\cdot)\) \(\chi_{6004}(359,\cdot)\) \(\chi_{6004}(423,\cdot)\) \(\chi_{6004}(443,\cdot)\) \(\chi_{6004}(503,\cdot)\) \(\chi_{6004}(511,\cdot)\) \(\chi_{6004}(587,\cdot)\) \(\chi_{6004}(671,\cdot)\) \(\chi_{6004}(707,\cdot)\) \(\chi_{6004}(739,\cdot)\) \(\chi_{6004}(1087,\cdot)\) \(\chi_{6004}(1183,\cdot)\) \(\chi_{6004}(1251,\cdot)\) \(\chi_{6004}(1327,\cdot)\) \(\chi_{6004}(1339,\cdot)\) \(\chi_{6004}(1391,\cdot)\) \(\chi_{6004}(1499,\cdot)\) \(\chi_{6004}(1507,\cdot)\) \(\chi_{6004}(1555,\cdot)\) \(\chi_{6004}(1619,\cdot)\) \(\chi_{6004}(1639,\cdot)\) \(\chi_{6004}(1719,\cdot)\) \(\chi_{6004}(1887,\cdot)\) \(\chi_{6004}(1955,\cdot)\) \(\chi_{6004}(2023,\cdot)\) ...

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: $\Q(\zeta_{117})$
Fixed field: Number field defined by a degree 234 polynomial (not computed)

Values on generators

\((3003,2529,3953)\) → \((-1,e\left(\frac{2}{9}\right),e\left(\frac{37}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 6004 }(35, a) \) \(1\)\(1\)\(e\left(\frac{101}{117}\right)\)\(e\left(\frac{113}{117}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{85}{117}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{28}{117}\right)\)\(e\left(\frac{97}{117}\right)\)\(e\left(\frac{43}{234}\right)\)\(e\left(\frac{98}{117}\right)\)\(e\left(\frac{5}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6004 }(35,a) \;\) at \(\;a = \) e.g. 2