Properties

Label 4730.67
Modulus $4730$
Conductor $215$
Order $84$
Real no
Primitive no
Minimal yes
Parity odd

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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([21,0,80]))
 
pari: [g,chi] = znchar(Mod(67,4730))
 

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 \(\chi_{215}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.cy

\(\chi_{4730}(23,\cdot)\) \(\chi_{4730}(67,\cdot)\) \(\chi_{4730}(353,\cdot)\) \(\chi_{4730}(397,\cdot)\) \(\chi_{4730}(573,\cdot)\) \(\chi_{4730}(617,\cdot)\) \(\chi_{4730}(683,\cdot)\) \(\chi_{4730}(1013,\cdot)\) \(\chi_{4730}(1057,\cdot)\) \(\chi_{4730}(1343,\cdot)\) \(\chi_{4730}(1563,\cdot)\) \(\chi_{4730}(1717,\cdot)\) \(\chi_{4730}(2003,\cdot)\) \(\chi_{4730}(2267,\cdot)\) \(\chi_{4730}(2597,\cdot)\) \(\chi_{4730}(2663,\cdot)\) \(\chi_{4730}(3213,\cdot)\) \(\chi_{4730}(3367,\cdot)\) \(\chi_{4730}(3543,\cdot)\) \(\chi_{4730}(3807,\cdot)\) \(\chi_{4730}(4137,\cdot)\) \(\chi_{4730}(4313,\cdot)\) \(\chi_{4730}(4357,\cdot)\) \(\chi_{4730}(4467,\cdot)\)

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

\((947,431,1981)\) → \((i,1,e\left(\frac{20}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(67, a) \) \(-1\)\(1\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{83}{84}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{23}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(67,a) \;\) at \(\;a = \) e.g. 2