Properties

Label 6004.65
Modulus $6004$
Conductor $1501$
Order $78$
Real no
Primitive no
Minimal yes
Parity odd

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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([0,13,18]))
 
pari: [g,chi] = znchar(Mod(65,6004))
 

Basic properties

Modulus: \(6004\)
Conductor: \(1501\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1501}(65,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6004.df

\(\chi_{6004}(65,\cdot)\) \(\chi_{6004}(141,\cdot)\) \(\chi_{6004}(749,\cdot)\) \(\chi_{6004}(1285,\cdot)\) \(\chi_{6004}(1361,\cdot)\) \(\chi_{6004}(1509,\cdot)\) \(\chi_{6004}(2121,\cdot)\) \(\chi_{6004}(2197,\cdot)\) \(\chi_{6004}(2501,\cdot)\) \(\chi_{6004}(2653,\cdot)\) \(\chi_{6004}(3181,\cdot)\) \(\chi_{6004}(3257,\cdot)\) \(\chi_{6004}(3261,\cdot)\) \(\chi_{6004}(3565,\cdot)\) \(\chi_{6004}(4017,\cdot)\) \(\chi_{6004}(4093,\cdot)\) \(\chi_{6004}(4173,\cdot)\) \(\chi_{6004}(4249,\cdot)\) \(\chi_{6004}(4397,\cdot)\) \(\chi_{6004}(4549,\cdot)\) \(\chi_{6004}(4857,\cdot)\) \(\chi_{6004}(5157,\cdot)\) \(\chi_{6004}(5461,\cdot)\) \(\chi_{6004}(5617,\cdot)\)

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

\((3003,2529,3953)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 6004 }(65, a) \) \(-1\)\(1\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{29}{78}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6004 }(65,a) \;\) at \(\;a = \) e.g. 2