Properties

Label 3751.251
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([18,1]))
 
pari: [g,chi] = znchar(Mod(251,3751))
 

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 \(\chi_{341}(251,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3751.bw

\(\chi_{3751}(251,\cdot)\) \(\chi_{3751}(269,\cdot)\) \(\chi_{3751}(632,\cdot)\) \(\chi_{3751}(1098,\cdot)\) \(\chi_{3751}(1412,\cdot)\) \(\chi_{3751}(2181,\cdot)\) \(\chi_{3751}(2907,\cdot)\) \(\chi_{3751}(3711,\cdot)\)

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: 30.0.174949846185675996887411683488270911769488970043898517884513025302911.3

Values on generators

\((2543,2421)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 3751 }(251, a) \) \(-1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{19}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3751 }(251,a) \;\) at \(\;a = \) e.g. 2