Properties

Label 9900.3103
Modulus $9900$
Conductor $900$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9900, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,40,21,0]))
 
pari: [g,chi] = znchar(Mod(3103,9900))
 

Basic properties

Modulus: \(9900\)
Conductor: \(900\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{900}(403,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9900.lk

\(\chi_{9900}(67,\cdot)\) \(\chi_{9900}(463,\cdot)\) \(\chi_{9900}(727,\cdot)\) \(\chi_{9900}(1123,\cdot)\) \(\chi_{9900}(2047,\cdot)\) \(\chi_{9900}(3103,\cdot)\) \(\chi_{9900}(4027,\cdot)\) \(\chi_{9900}(4423,\cdot)\) \(\chi_{9900}(4687,\cdot)\) \(\chi_{9900}(5083,\cdot)\) \(\chi_{9900}(6403,\cdot)\) \(\chi_{9900}(6667,\cdot)\) \(\chi_{9900}(7063,\cdot)\) \(\chi_{9900}(7987,\cdot)\) \(\chi_{9900}(8383,\cdot)\) \(\chi_{9900}(8647,\cdot)\)

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

\((4951,5501,2377,4501)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{7}{20}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9900 }(3103, a) \) \(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9900 }(3103,a) \;\) at \(\;a = \) e.g. 2