Properties

Label 8640.119
Modulus $8640$
Conductor $4320$
Order $72$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8640, base_ring=CyclotomicField(72))
 
M = H._module
 
chi = DirichletCharacter(H, M([36,27,52,36]))
 
pari: [g,chi] = znchar(Mod(119,8640))
 

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 \(\chi_{4320}(2819,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8640.ho

\(\chi_{8640}(119,\cdot)\) \(\chi_{8640}(599,\cdot)\) \(\chi_{8640}(839,\cdot)\) \(\chi_{8640}(1319,\cdot)\) \(\chi_{8640}(1559,\cdot)\) \(\chi_{8640}(2039,\cdot)\) \(\chi_{8640}(2279,\cdot)\) \(\chi_{8640}(2759,\cdot)\) \(\chi_{8640}(2999,\cdot)\) \(\chi_{8640}(3479,\cdot)\) \(\chi_{8640}(3719,\cdot)\) \(\chi_{8640}(4199,\cdot)\) \(\chi_{8640}(4439,\cdot)\) \(\chi_{8640}(4919,\cdot)\) \(\chi_{8640}(5159,\cdot)\) \(\chi_{8640}(5639,\cdot)\) \(\chi_{8640}(5879,\cdot)\) \(\chi_{8640}(6359,\cdot)\) \(\chi_{8640}(6599,\cdot)\) \(\chi_{8640}(7079,\cdot)\) \(\chi_{8640}(7319,\cdot)\) \(\chi_{8640}(7799,\cdot)\) \(\chi_{8640}(8039,\cdot)\) \(\chi_{8640}(8519,\cdot)\)

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

\((2431,3781,6401,3457)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{13}{18}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8640 }(119, a) \) \(1\)\(1\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{55}{72}\right)\)\(e\left(\frac{65}{72}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{61}{72}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{19}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8640 }(119,a) \;\) at \(\;a = \) e.g. 2