Properties

Label 4730.61
Modulus $4730$
Conductor $473$
Order $210$
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(4730, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,189,145]))
 
pari: [g,chi] = znchar(Mod(61,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.dn

\(\chi_{4730}(61,\cdot)\) \(\chi_{4730}(261,\cdot)\) \(\chi_{4730}(321,\cdot)\) \(\chi_{4730}(491,\cdot)\) \(\chi_{4730}(501,\cdot)\) \(\chi_{4730}(721,\cdot)\) \(\chi_{4730}(761,\cdot)\) \(\chi_{4730}(921,\cdot)\) \(\chi_{4730}(931,\cdot)\) \(\chi_{4730}(1051,\cdot)\) \(\chi_{4730}(1151,\cdot)\) \(\chi_{4730}(1361,\cdot)\) \(\chi_{4730}(1381,\cdot)\) \(\chi_{4730}(1481,\cdot)\) \(\chi_{4730}(1491,\cdot)\) \(\chi_{4730}(1531,\cdot)\) \(\chi_{4730}(1581,\cdot)\) \(\chi_{4730}(1711,\cdot)\) \(\chi_{4730}(1811,\cdot)\) \(\chi_{4730}(1861,\cdot)\) \(\chi_{4730}(1911,\cdot)\) \(\chi_{4730}(1921,\cdot)\) \(\chi_{4730}(2041,\cdot)\) \(\chi_{4730}(2141,\cdot)\) \(\chi_{4730}(2241,\cdot)\) \(\chi_{4730}(2351,\cdot)\) \(\chi_{4730}(2411,\cdot)\) \(\chi_{4730}(2471,\cdot)\) \(\chi_{4730}(2481,\cdot)\) \(\chi_{4730}(2571,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((947,431,1981)\) → \((1,e\left(\frac{9}{10}\right),e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(61, a) \) \(1\)\(1\)\(e\left(\frac{187}{210}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{209}{210}\right)\)\(e\left(\frac{71}{210}\right)\)\(e\left(\frac{86}{105}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{47}{70}\right)\)\(e\left(\frac{64}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(61,a) \;\) at \(\;a = \) e.g. 2