from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,16]))
pari: [g,chi] = znchar(Mod(7,1296))
Basic properties
| Modulus: | \(1296\) | |
| Conductor: | \(648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{648}(331,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1296.br
\(\chi_{1296}(7,\cdot)\) \(\chi_{1296}(103,\cdot)\) \(\chi_{1296}(151,\cdot)\) \(\chi_{1296}(247,\cdot)\) \(\chi_{1296}(295,\cdot)\) \(\chi_{1296}(391,\cdot)\) \(\chi_{1296}(439,\cdot)\) \(\chi_{1296}(535,\cdot)\) \(\chi_{1296}(583,\cdot)\) \(\chi_{1296}(679,\cdot)\) \(\chi_{1296}(727,\cdot)\) \(\chi_{1296}(823,\cdot)\) \(\chi_{1296}(871,\cdot)\) \(\chi_{1296}(967,\cdot)\) \(\chi_{1296}(1015,\cdot)\) \(\chi_{1296}(1111,\cdot)\) \(\chi_{1296}(1159,\cdot)\) \(\chi_{1296}(1255,\cdot)\)
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
\((1135,325,1217)\) → \((-1,-1,e\left(\frac{8}{27}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1296 }(7, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{23}{54}\right)\) |
sage: chi.jacobi_sum(n)