Properties

Label 8042.37
Modulus $8042$
Conductor $4021$
Order $15$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8042, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([16]))
 
pari: [g,chi] = znchar(Mod(37,8042))
 

Basic properties

Modulus: \(8042\)
Conductor: \(4021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4021}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8042.i

\(\chi_{8042}(37,\cdot)\) \(\chi_{8042}(375,\cdot)\) \(\chi_{8042}(1369,\cdot)\) \(\chi_{8042}(3695,\cdot)\) \(\chi_{8042}(3911,\cdot)\) \(\chi_{8042}(5297,\cdot)\) \(\chi_{8042}(5751,\cdot)\) \(\chi_{8042}(7713,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\(4023\) → \(e\left(\frac{8}{15}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8042 }(37, a) \) \(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8042 }(37,a) \;\) at \(\;a = \) e.g. 2