from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6480, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,29,27]))
pari: [g,chi] = znchar(Mod(239,6480))
Basic properties
Modulus: | \(6480\) | |
Conductor: | \(1620\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1620}(239,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6480.eu
\(\chi_{6480}(239,\cdot)\) \(\chi_{6480}(479,\cdot)\) \(\chi_{6480}(959,\cdot)\) \(\chi_{6480}(1199,\cdot)\) \(\chi_{6480}(1679,\cdot)\) \(\chi_{6480}(1919,\cdot)\) \(\chi_{6480}(2399,\cdot)\) \(\chi_{6480}(2639,\cdot)\) \(\chi_{6480}(3119,\cdot)\) \(\chi_{6480}(3359,\cdot)\) \(\chi_{6480}(3839,\cdot)\) \(\chi_{6480}(4079,\cdot)\) \(\chi_{6480}(4559,\cdot)\) \(\chi_{6480}(4799,\cdot)\) \(\chi_{6480}(5279,\cdot)\) \(\chi_{6480}(5519,\cdot)\) \(\chi_{6480}(5999,\cdot)\) \(\chi_{6480}(6239,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((2431,1621,6401,1297)\) → \((-1,1,e\left(\frac{29}{54}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 6480 }(239, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{25}{54}\right)\) |
sage: chi.jacobi_sum(n)