from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([41,27]))
pari: [g,chi] = znchar(Mod(59,810))
Basic properties
| Modulus: | \(810\) | |
| Conductor: | \(405\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{405}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 810.t
\(\chi_{810}(29,\cdot)\) \(\chi_{810}(59,\cdot)\) \(\chi_{810}(119,\cdot)\) \(\chi_{810}(149,\cdot)\) \(\chi_{810}(209,\cdot)\) \(\chi_{810}(239,\cdot)\) \(\chi_{810}(299,\cdot)\) \(\chi_{810}(329,\cdot)\) \(\chi_{810}(389,\cdot)\) \(\chi_{810}(419,\cdot)\) \(\chi_{810}(479,\cdot)\) \(\chi_{810}(509,\cdot)\) \(\chi_{810}(569,\cdot)\) \(\chi_{810}(599,\cdot)\) \(\chi_{810}(659,\cdot)\) \(\chi_{810}(689,\cdot)\) \(\chi_{810}(749,\cdot)\) \(\chi_{810}(779,\cdot)\)
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
\((731,487)\) → \((e\left(\frac{41}{54}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 810 }(59, a) \) | \(-1\) | \(1\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{54}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)