Properties

Label 4730.3587
Modulus $4730$
Conductor $215$
Order $84$
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(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,0,58]))
 
pari: [g,chi] = znchar(Mod(3587,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(215\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{215}(147,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.da

\(\chi_{4730}(177,\cdot)\) \(\chi_{4730}(243,\cdot)\) \(\chi_{4730}(287,\cdot)\) \(\chi_{4730}(463,\cdot)\) \(\chi_{4730}(507,\cdot)\) \(\chi_{4730}(793,\cdot)\) \(\chi_{4730}(837,\cdot)\) \(\chi_{4730}(1123,\cdot)\) \(\chi_{4730}(1233,\cdot)\) \(\chi_{4730}(1277,\cdot)\) \(\chi_{4730}(1453,\cdot)\) \(\chi_{4730}(1783,\cdot)\) \(\chi_{4730}(2047,\cdot)\) \(\chi_{4730}(2223,\cdot)\) \(\chi_{4730}(2377,\cdot)\) \(\chi_{4730}(2927,\cdot)\) \(\chi_{4730}(2993,\cdot)\) \(\chi_{4730}(3323,\cdot)\) \(\chi_{4730}(3587,\cdot)\) \(\chi_{4730}(3873,\cdot)\) \(\chi_{4730}(4027,\cdot)\) \(\chi_{4730}(4247,\cdot)\) \(\chi_{4730}(4533,\cdot)\) \(\chi_{4730}(4577,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(3587, a) \) \(1\)\(1\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{41}{84}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{17}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(3587,a) \;\) at \(\;a = \) e.g. 2