Properties

Label 4730.549
Modulus $4730$
Conductor $2365$
Order $42$
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(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,21,31]))
 
pari: [g,chi] = znchar(Mod(549,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(2365\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2365}(549,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.cm

\(\chi_{4730}(329,\cdot)\) \(\chi_{4730}(549,\cdot)\) \(\chi_{4730}(879,\cdot)\) \(\chi_{4730}(1209,\cdot)\) \(\chi_{4730}(1319,\cdot)\) \(\chi_{4730}(1539,\cdot)\) \(\chi_{4730}(1869,\cdot)\) \(\chi_{4730}(2309,\cdot)\) \(\chi_{4730}(3079,\cdot)\) \(\chi_{4730}(3409,\cdot)\) \(\chi_{4730}(3959,\cdot)\) \(\chi_{4730}(4619,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((947,431,1981)\) → \((-1,-1,e\left(\frac{31}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(549, a) \) \(1\)\(1\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{16}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(549,a) \;\) at \(\;a = \) e.g. 2