Properties

Label 9900.607
Modulus $9900$
Conductor $1980$
Order $60$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9900, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,20,15,6]))
 
pari: [g,chi] = znchar(Mod(607,9900))
 

Basic properties

Modulus: \(9900\)
Conductor: \(1980\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1980}(607,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9900.mm

\(\chi_{9900}(7,\cdot)\) \(\chi_{9900}(607,\cdot)\) \(\chi_{9900}(943,\cdot)\) \(\chi_{9900}(1843,\cdot)\) \(\chi_{9900}(3307,\cdot)\) \(\chi_{9900}(3643,\cdot)\) \(\chi_{9900}(4243,\cdot)\) \(\chi_{9900}(4507,\cdot)\) \(\chi_{9900}(5143,\cdot)\) \(\chi_{9900}(5407,\cdot)\) \(\chi_{9900}(6343,\cdot)\) \(\chi_{9900}(6943,\cdot)\) \(\chi_{9900}(7207,\cdot)\) \(\chi_{9900}(7807,\cdot)\) \(\chi_{9900}(8707,\cdot)\) \(\chi_{9900}(9643,\cdot)\)

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

\((4951,5501,2377,4501)\) → \((-1,e\left(\frac{1}{3}\right),i,e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9900 }(607, a) \) \(-1\)\(1\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9900 }(607,a) \;\) at \(\;a = \) e.g. 2