Properties

Label 8640.137
Modulus $8640$
Conductor $4320$
Order $72$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8640, base_ring=CyclotomicField(72))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,27,4,18]))
 
pari: [g,chi] = znchar(Mod(137,8640))
 

Basic properties

Modulus: \(8640\)
Conductor: \(4320\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(72\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4320}(1757,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8640.hg

\(\chi_{8640}(137,\cdot)\) \(\chi_{8640}(473,\cdot)\) \(\chi_{8640}(617,\cdot)\) \(\chi_{8640}(1433,\cdot)\) \(\chi_{8640}(1577,\cdot)\) \(\chi_{8640}(1913,\cdot)\) \(\chi_{8640}(2057,\cdot)\) \(\chi_{8640}(2873,\cdot)\) \(\chi_{8640}(3017,\cdot)\) \(\chi_{8640}(3353,\cdot)\) \(\chi_{8640}(3497,\cdot)\) \(\chi_{8640}(4313,\cdot)\) \(\chi_{8640}(4457,\cdot)\) \(\chi_{8640}(4793,\cdot)\) \(\chi_{8640}(4937,\cdot)\) \(\chi_{8640}(5753,\cdot)\) \(\chi_{8640}(5897,\cdot)\) \(\chi_{8640}(6233,\cdot)\) \(\chi_{8640}(6377,\cdot)\) \(\chi_{8640}(7193,\cdot)\) \(\chi_{8640}(7337,\cdot)\) \(\chi_{8640}(7673,\cdot)\) \(\chi_{8640}(7817,\cdot)\) \(\chi_{8640}(8633,\cdot)\)

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: $\Q(\zeta_{72})$
Fixed field: Number field defined by a degree 72 polynomial

Values on generators

\((2431,3781,6401,3457)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{1}{18}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8640 }(137, a) \) \(1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{43}{72}\right)\)\(e\left(\frac{59}{72}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{49}{72}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{7}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8640 }(137,a) \;\) at \(\;a = \) e.g. 2