from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,7]))
pari: [g,chi] = znchar(Mod(47,1296))
Basic properties
| Modulus: | \(1296\) | |
| Conductor: | \(324\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{324}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1296.bq
\(\chi_{1296}(47,\cdot)\) \(\chi_{1296}(95,\cdot)\) \(\chi_{1296}(191,\cdot)\) \(\chi_{1296}(239,\cdot)\) \(\chi_{1296}(335,\cdot)\) \(\chi_{1296}(383,\cdot)\) \(\chi_{1296}(479,\cdot)\) \(\chi_{1296}(527,\cdot)\) \(\chi_{1296}(623,\cdot)\) \(\chi_{1296}(671,\cdot)\) \(\chi_{1296}(767,\cdot)\) \(\chi_{1296}(815,\cdot)\) \(\chi_{1296}(911,\cdot)\) \(\chi_{1296}(959,\cdot)\) \(\chi_{1296}(1055,\cdot)\) \(\chi_{1296}(1103,\cdot)\) \(\chi_{1296}(1199,\cdot)\) \(\chi_{1296}(1247,\cdot)\)
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
\((1135,325,1217)\) → \((-1,1,e\left(\frac{7}{54}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1296 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{5}{54}\right)\) |
sage: chi.jacobi_sum(n)