Properties

Label 4004.59
Modulus $4004$
Conductor $4004$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,10,12,55]))
 
pari: [g,chi] = znchar(Mod(59,4004))
 

Basic properties

Modulus: \(4004\)
Conductor: \(4004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4004.ir

\(\chi_{4004}(59,\cdot)\) \(\chi_{4004}(423,\cdot)\) \(\chi_{4004}(691,\cdot)\) \(\chi_{4004}(955,\cdot)\) \(\chi_{4004}(999,\cdot)\) \(\chi_{4004}(1879,\cdot)\) \(\chi_{4004}(2511,\cdot)\) \(\chi_{4004}(2775,\cdot)\) \(\chi_{4004}(2819,\cdot)\) \(\chi_{4004}(2875,\cdot)\) \(\chi_{4004}(3139,\cdot)\) \(\chi_{4004}(3183,\cdot)\) \(\chi_{4004}(3239,\cdot)\) \(\chi_{4004}(3503,\cdot)\) \(\chi_{4004}(3547,\cdot)\) \(\chi_{4004}(3699,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((2003,3433,365,925)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{5}\right),e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)\(29\)
\( \chi_{ 4004 }(59, a) \) \(-1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{31}{60}\right)\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4004 }(59,a) \;\) at \(\;a = \) e.g. 2