Properties

Label 3822.185
Modulus $3822$
Conductor $1911$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,19,14]))
 
pari: [g,chi] = znchar(Mod(185,3822))
 

Basic properties

Modulus: \(3822\)
Conductor: \(1911\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1911}(185,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3822.dv

\(\chi_{3822}(185,\cdot)\) \(\chi_{3822}(425,\cdot)\) \(\chi_{3822}(731,\cdot)\) \(\chi_{3822}(971,\cdot)\) \(\chi_{3822}(1277,\cdot)\) \(\chi_{3822}(1517,\cdot)\) \(\chi_{3822}(1823,\cdot)\) \(\chi_{3822}(2063,\cdot)\) \(\chi_{3822}(2369,\cdot)\) \(\chi_{3822}(2609,\cdot)\) \(\chi_{3822}(2915,\cdot)\) \(\chi_{3822}(3701,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((2549,3433,1471)\) → \((-1,e\left(\frac{19}{42}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3822 }(185, a) \) \(1\)\(1\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{10}{21}\right)\)\(-1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{13}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3822 }(185,a) \;\) at \(\;a = \) e.g. 2