Properties

Label 4730.361
Modulus $4730$
Conductor $473$
Order $105$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4730, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,126,190]))
 
pari: [g,chi] = znchar(Mod(361,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(361,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.dc

\(\chi_{4730}(31,\cdot)\) \(\chi_{4730}(81,\cdot)\) \(\chi_{4730}(181,\cdot)\) \(\chi_{4730}(311,\cdot)\) \(\chi_{4730}(361,\cdot)\) \(\chi_{4730}(401,\cdot)\) \(\chi_{4730}(411,\cdot)\) \(\chi_{4730}(511,\cdot)\) \(\chi_{4730}(531,\cdot)\) \(\chi_{4730}(741,\cdot)\) \(\chi_{4730}(841,\cdot)\) \(\chi_{4730}(961,\cdot)\) \(\chi_{4730}(971,\cdot)\) \(\chi_{4730}(1131,\cdot)\) \(\chi_{4730}(1171,\cdot)\) \(\chi_{4730}(1391,\cdot)\) \(\chi_{4730}(1401,\cdot)\) \(\chi_{4730}(1571,\cdot)\) \(\chi_{4730}(1631,\cdot)\) \(\chi_{4730}(1831,\cdot)\) \(\chi_{4730}(1901,\cdot)\) \(\chi_{4730}(2061,\cdot)\) \(\chi_{4730}(2121,\cdot)\) \(\chi_{4730}(2181,\cdot)\) \(\chi_{4730}(2231,\cdot)\) \(\chi_{4730}(2491,\cdot)\) \(\chi_{4730}(2511,\cdot)\) \(\chi_{4730}(2561,\cdot)\) \(\chi_{4730}(2611,\cdot)\) \(\chi_{4730}(2891,\cdot)\) ...

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 105 polynomial (not computed)

Values on generators

\((947,431,1981)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{19}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(361, a) \) \(1\)\(1\)\(e\left(\frac{74}{105}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{43}{105}\right)\)\(e\left(\frac{58}{105}\right)\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{31}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(361,a) \;\) at \(\;a = \) e.g. 2