Properties

Label 9900.5663
Modulus $9900$
Conductor $9900$
Order $60$
Real no
Primitive yes
Minimal yes
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([30,10,57,36]))
 
pari: [g,chi] = znchar(Mod(5663,9900))
 

Basic properties

Modulus: \(9900\)
Conductor: \(9900\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9900.lt

\(\chi_{9900}(47,\cdot)\) \(\chi_{9900}(983,\cdot)\) \(\chi_{9900}(1103,\cdot)\) \(\chi_{9900}(1127,\cdot)\) \(\chi_{9900}(1523,\cdot)\) \(\chi_{9900}(2363,\cdot)\) \(\chi_{9900}(2567,\cdot)\) \(\chi_{9900}(3287,\cdot)\) \(\chi_{9900}(4403,\cdot)\) \(\chi_{9900}(5663,\cdot)\) \(\chi_{9900}(6647,\cdot)\) \(\chi_{9900}(7583,\cdot)\) \(\chi_{9900}(7727,\cdot)\) \(\chi_{9900}(8123,\cdot)\) \(\chi_{9900}(9167,\cdot)\) \(\chi_{9900}(9887,\cdot)\)

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

\((4951,5501,2377,4501)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{19}{20}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9900 }(5663, a) \) \(-1\)\(1\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{59}{60}\right)\)\(i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(-i\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9900 }(5663,a) \;\) at \(\;a = \) e.g. 2