from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([20,0]))
pari: [g,chi] = znchar(Mod(31,810))
Basic properties
| Modulus: | \(810\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 810.q
\(\chi_{810}(31,\cdot)\) \(\chi_{810}(61,\cdot)\) \(\chi_{810}(121,\cdot)\) \(\chi_{810}(151,\cdot)\) \(\chi_{810}(211,\cdot)\) \(\chi_{810}(241,\cdot)\) \(\chi_{810}(301,\cdot)\) \(\chi_{810}(331,\cdot)\) \(\chi_{810}(391,\cdot)\) \(\chi_{810}(421,\cdot)\) \(\chi_{810}(481,\cdot)\) \(\chi_{810}(511,\cdot)\) \(\chi_{810}(571,\cdot)\) \(\chi_{810}(601,\cdot)\) \(\chi_{810}(661,\cdot)\) \(\chi_{810}(691,\cdot)\) \(\chi_{810}(751,\cdot)\) \(\chi_{810}(781,\cdot)\)
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 27 polynomial |
Values on generators
\((731,487)\) → \((e\left(\frac{10}{27}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 810 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) |
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)