from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,33]))
pari: [g,chi] = znchar(Mod(523,675))
Basic properties
| Modulus: | \(675\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(148,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 675.be
\(\chi_{675}(37,\cdot)\) \(\chi_{675}(73,\cdot)\) \(\chi_{675}(127,\cdot)\) \(\chi_{675}(172,\cdot)\) \(\chi_{675}(208,\cdot)\) \(\chi_{675}(253,\cdot)\) \(\chi_{675}(262,\cdot)\) \(\chi_{675}(388,\cdot)\) \(\chi_{675}(397,\cdot)\) \(\chi_{675}(442,\cdot)\) \(\chi_{675}(478,\cdot)\) \(\chi_{675}(523,\cdot)\) \(\chi_{675}(577,\cdot)\) \(\chi_{675}(613,\cdot)\) \(\chi_{675}(658,\cdot)\) \(\chi_{675}(667,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((326,352)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 675 }(523, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
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)