Properties

Label 4730.1731
Modulus $4730$
Conductor $473$
Order $35$
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(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14,50]))
 
pari: [g,chi] = znchar(Mod(1731,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(312,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.cf

\(\chi_{4730}(471,\cdot)\) \(\chi_{4730}(1301,\cdot)\) \(\chi_{4730}(1411,\cdot)\) \(\chi_{4730}(1521,\cdot)\) \(\chi_{4730}(1681,\cdot)\) \(\chi_{4730}(1731,\cdot)\) \(\chi_{4730}(1741,\cdot)\) \(\chi_{4730}(1841,\cdot)\) \(\chi_{4730}(1951,\cdot)\) \(\chi_{4730}(2161,\cdot)\) \(\chi_{4730}(2171,\cdot)\) \(\chi_{4730}(2271,\cdot)\) \(\chi_{4730}(2381,\cdot)\) \(\chi_{4730}(2601,\cdot)\) \(\chi_{4730}(2621,\cdot)\) \(\chi_{4730}(3051,\cdot)\) \(\chi_{4730}(3481,\cdot)\) \(\chi_{4730}(3831,\cdot)\) \(\chi_{4730}(3881,\cdot)\) \(\chi_{4730}(3991,\cdot)\) \(\chi_{4730}(4101,\cdot)\) \(\chi_{4730}(4261,\cdot)\) \(\chi_{4730}(4321,\cdot)\) \(\chi_{4730}(4691,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 35 polynomial

Values on generators

\((947,431,1981)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{5}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(1731, a) \) \(1\)\(1\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{24}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(1731,a) \;\) at \(\;a = \) e.g. 2