Properties

Label 3751.483
Modulus $3751$
Conductor $341$
Order $30$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3751, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,26]))
 
pari: [g,chi] = znchar(Mod(483,3751))
 

Basic properties

Modulus: \(3751\)
Conductor: \(341\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{341}(142,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3751.cc

\(\chi_{3751}(483,\cdot)\) \(\chi_{3751}(846,\cdot)\) \(\chi_{3751}(1330,\cdot)\) \(\chi_{3751}(1693,\cdot)\) \(\chi_{3751}(2056,\cdot)\) \(\chi_{3751}(2177,\cdot)\) \(\chi_{3751}(2903,\cdot)\) \(\chi_{3751}(3145,\cdot)\)

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 30 polynomial

Values on generators

\((2543,2421)\) → \((-1,e\left(\frac{13}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 3751 }(483, a) \) \(-1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3751 }(483,a) \;\) at \(\;a = \) e.g. 2