from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(235, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,38]))
pari: [g,chi] = znchar(Mod(6,235))
Basic properties
| Modulus: | \(235\) | |
| Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{47}(6,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 235.g
\(\chi_{235}(6,\cdot)\) \(\chi_{235}(16,\cdot)\) \(\chi_{235}(21,\cdot)\) \(\chi_{235}(36,\cdot)\) \(\chi_{235}(51,\cdot)\) \(\chi_{235}(56,\cdot)\) \(\chi_{235}(61,\cdot)\) \(\chi_{235}(71,\cdot)\) \(\chi_{235}(81,\cdot)\) \(\chi_{235}(96,\cdot)\) \(\chi_{235}(101,\cdot)\) \(\chi_{235}(106,\cdot)\) \(\chi_{235}(111,\cdot)\) \(\chi_{235}(121,\cdot)\) \(\chi_{235}(126,\cdot)\) \(\chi_{235}(131,\cdot)\) \(\chi_{235}(136,\cdot)\) \(\chi_{235}(166,\cdot)\) \(\chi_{235}(191,\cdot)\) \(\chi_{235}(196,\cdot)\) \(\chi_{235}(206,\cdot)\) \(\chi_{235}(216,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\((142,146)\) → \((1,e\left(\frac{19}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 235 }(6, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) |
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)