Properties

Label 3969.25
Modulus $3969$
Conductor $3969$
Order $189$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3969, base_ring=CyclotomicField(378))
 
M = H._module
 
chi = DirichletCharacter(H, M([322,144]))
 
pari: [g,chi] = znchar(Mod(25,3969))
 

Basic properties

Modulus: \(3969\)
Conductor: \(3969\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(189\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3969.cy

\(\chi_{3969}(25,\cdot)\) \(\chi_{3969}(58,\cdot)\) \(\chi_{3969}(88,\cdot)\) \(\chi_{3969}(121,\cdot)\) \(\chi_{3969}(151,\cdot)\) \(\chi_{3969}(184,\cdot)\) \(\chi_{3969}(247,\cdot)\) \(\chi_{3969}(277,\cdot)\) \(\chi_{3969}(310,\cdot)\) \(\chi_{3969}(340,\cdot)\) \(\chi_{3969}(403,\cdot)\) \(\chi_{3969}(436,\cdot)\) \(\chi_{3969}(466,\cdot)\) \(\chi_{3969}(499,\cdot)\) \(\chi_{3969}(529,\cdot)\) \(\chi_{3969}(562,\cdot)\) \(\chi_{3969}(592,\cdot)\) \(\chi_{3969}(625,\cdot)\) \(\chi_{3969}(688,\cdot)\) \(\chi_{3969}(718,\cdot)\) \(\chi_{3969}(751,\cdot)\) \(\chi_{3969}(781,\cdot)\) \(\chi_{3969}(844,\cdot)\) \(\chi_{3969}(877,\cdot)\) \(\chi_{3969}(907,\cdot)\) \(\chi_{3969}(940,\cdot)\) \(\chi_{3969}(970,\cdot)\) \(\chi_{3969}(1003,\cdot)\) \(\chi_{3969}(1033,\cdot)\) \(\chi_{3969}(1066,\cdot)\) ...

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: $\Q(\zeta_{189})$
Fixed field: Number field defined by a degree 189 polynomial (not computed)

Values on generators

\((2108,3727)\) → \((e\left(\frac{23}{27}\right),e\left(\frac{8}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3969 }(25, a) \) \(1\)\(1\)\(e\left(\frac{143}{189}\right)\)\(e\left(\frac{97}{189}\right)\)\(e\left(\frac{121}{189}\right)\)\(e\left(\frac{17}{63}\right)\)\(e\left(\frac{25}{63}\right)\)\(e\left(\frac{59}{189}\right)\)\(e\left(\frac{73}{189}\right)\)\(e\left(\frac{5}{189}\right)\)\(e\left(\frac{40}{63}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3969 }(25,a) \;\) at \(\;a = \) e.g. 2