Properties

Label 6027.374
Modulus $6027$
Conductor $861$
Order $60$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,10,33]))
 
pari: [g,chi] = znchar(Mod(374,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(861\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{861}(374,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.df

\(\chi_{6027}(80,\cdot)\) \(\chi_{6027}(374,\cdot)\) \(\chi_{6027}(815,\cdot)\) \(\chi_{6027}(1109,\cdot)\) \(\chi_{6027}(1538,\cdot)\) \(\chi_{6027}(1550,\cdot)\) \(\chi_{6027}(2714,\cdot)\) \(\chi_{6027}(2726,\cdot)\) \(\chi_{6027}(3155,\cdot)\) \(\chi_{6027}(3449,\cdot)\) \(\chi_{6027}(3890,\cdot)\) \(\chi_{6027}(4184,\cdot)\) \(\chi_{6027}(4490,\cdot)\) \(\chi_{6027}(4625,\cdot)\) \(\chi_{6027}(5666,\cdot)\) \(\chi_{6027}(5801,\cdot)\)

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

\((4019,493,2794)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{11}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(374, a) \) \(1\)\(1\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{47}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(374,a) \;\) at \(\;a = \) e.g. 2