Properties

Label 3900.89
Modulus $3900$
Conductor $975$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3900, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,30,18,35]))
 
pari: [g,chi] = znchar(Mod(89,3900))
 

Basic properties

Modulus: \(3900\)
Conductor: \(975\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{975}(89,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3900.fo

\(\chi_{3900}(89,\cdot)\) \(\chi_{3900}(509,\cdot)\) \(\chi_{3900}(869,\cdot)\) \(\chi_{3900}(929,\cdot)\) \(\chi_{3900}(1229,\cdot)\) \(\chi_{3900}(1289,\cdot)\) \(\chi_{3900}(1709,\cdot)\) \(\chi_{3900}(2009,\cdot)\) \(\chi_{3900}(2069,\cdot)\) \(\chi_{3900}(2429,\cdot)\) \(\chi_{3900}(2489,\cdot)\) \(\chi_{3900}(2789,\cdot)\) \(\chi_{3900}(3209,\cdot)\) \(\chi_{3900}(3269,\cdot)\) \(\chi_{3900}(3569,\cdot)\) \(\chi_{3900}(3629,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1951,1301,3277,301)\) → \((1,-1,e\left(\frac{3}{10}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3900 }(89, a) \) \(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3900 }(89,a) \;\) at \(\;a = \) e.g. 2