from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([1,0]))
pari: [g,chi] = znchar(Mod(731,810))
Basic properties
Modulus: | \(810\) | |
Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{81}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 810.u
\(\chi_{810}(11,\cdot)\) \(\chi_{810}(41,\cdot)\) \(\chi_{810}(101,\cdot)\) \(\chi_{810}(131,\cdot)\) \(\chi_{810}(191,\cdot)\) \(\chi_{810}(221,\cdot)\) \(\chi_{810}(281,\cdot)\) \(\chi_{810}(311,\cdot)\) \(\chi_{810}(371,\cdot)\) \(\chi_{810}(401,\cdot)\) \(\chi_{810}(461,\cdot)\) \(\chi_{810}(491,\cdot)\) \(\chi_{810}(551,\cdot)\) \(\chi_{810}(581,\cdot)\) \(\chi_{810}(641,\cdot)\) \(\chi_{810}(671,\cdot)\) \(\chi_{810}(731,\cdot)\) \(\chi_{810}(761,\cdot)\)
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
\((731,487)\) → \((e\left(\frac{1}{54}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 810 }(731, a) \) | \(-1\) | \(1\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{53}{54}\right)\) |
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)