Properties

Label 6004.39
Modulus $6004$
Conductor $316$
Order $78$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6004, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,0,35]))
 
pari: [g,chi] = znchar(Mod(39,6004))
 

Basic properties

Modulus: \(6004\)
Conductor: \(316\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{316}(39,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6004.dt

\(\chi_{6004}(39,\cdot)\) \(\chi_{6004}(267,\cdot)\) \(\chi_{6004}(875,\cdot)\) \(\chi_{6004}(951,\cdot)\) \(\chi_{6004}(1255,\cdot)\) \(\chi_{6004}(1939,\cdot)\) \(\chi_{6004}(2091,\cdot)\) \(\chi_{6004}(2167,\cdot)\) \(\chi_{6004}(2319,\cdot)\) \(\chi_{6004}(2851,\cdot)\) \(\chi_{6004}(3079,\cdot)\) \(\chi_{6004}(3155,\cdot)\) \(\chi_{6004}(3307,\cdot)\) \(\chi_{6004}(3535,\cdot)\) \(\chi_{6004}(3687,\cdot)\) \(\chi_{6004}(3839,\cdot)\) \(\chi_{6004}(4143,\cdot)\) \(\chi_{6004}(4295,\cdot)\) \(\chi_{6004}(5131,\cdot)\) \(\chi_{6004}(5359,\cdot)\) \(\chi_{6004}(5435,\cdot)\) \(\chi_{6004}(5511,\cdot)\) \(\chi_{6004}(5663,\cdot)\) \(\chi_{6004}(5815,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((3003,2529,3953)\) → \((-1,1,e\left(\frac{35}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 6004 }(39, a) \) \(1\)\(1\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6004 }(39,a) \;\) at \(\;a = \) e.g. 2